10.5 Solve Exponential and Logarithmic Equations

Before you get started, take this readiness quiz.

Solve: $x^{2} = 16 .$
If you missed this problem, review the reference.

Solve: $x^{2} - 5 x + 6 = 0 .$
If you missed this problem, review the reference.

Solve: $x \left(x + 6\right) = 2 x + 5 .$
If you missed this problem, review the reference.

Solve: $2 \text{log}_{5} x = \text{log}_{5} 81 .$

Solve: $2 \text{log}_{3} x = \text{log}_{3} 36$

Solve: $3 \text{log} x = \text{log} 64$

Solve: $\text{log}_{3} x + \text{log}_{3} \left(x - 8\right) = 2 .$

Solve: $\text{log}_{2} x + \text{log}_{2} \left(x - 2\right) = 3$

Solve: $\text{log}_{2} x + \text{log}_{2} \left(x - 6\right) = 4$

Solve: $\text{log}_{4} \left(x + 6\right) - \text{log}_{4} \left(2 x + 5\right) = − \text{log}_{4} x .$

Solve: $\text{log} \left(x + 2\right) - \text{log} \left(4 x + 3\right) = − \text{log} x .$

Solve: $\text{log} \left(x - 2\right) - \text{log} \left(4 x + 16\right) = \text{log} \frac{1}{x} .$

Solve $5^{x} = 11 .$ Find the exact answer and then approximate it to three decimal places.

Solve $7^{x} = 43 .$ Find the exact answer and then approximate it to three decimal places.

Solve $8^{x} = 98 .$ Find the exact answer and then approximate it to three decimal places.

Solve $3 e^{x + 2} = 24 .$ Find the exact answer and then approximate it to three decimal places.

Solve $2 e^{x - 2} = 18 .$ Find the exact answer and then approximate it to three decimal places.

Solve $5 e^{2 x} = 25 .$ Find the exact answer and then approximate it to three decimal places.

Jermael’s parents put $10,000 in investments for his college expenses on his first birthday. They hope the investments will be worth $50,000 when he turns 18. If the interest compounds continuously, approximately what rate of growth will they need to achieve their goal?

Hector invests $\$ 10,000$ at age 21. He hopes the investments will be worth $\$ 150,000$ when he turns 50. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal?

Rachel invests $\$ 15,000$ at age 25. She hopes the investments will be worth $\$ 90,000$ when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Researchers recorded that a certain bacteria population grew from 100 to 300 in 3 hours. At this rate of growth, how many bacteria will there be 24 hours from the start of the experiment?

Researchers recorded that a certain bacteria population grew from 100 to 500 in 6 hours. At this rate of growth, how many bacteria will there be 24 hours from the start of the experiment?

Researchers recorded that a certain bacteria population declined from 700,000 to 400,000 in 5 hours after the administration of medication. At this rate of decay, how many bacteria will there be 24 hours from the start of the experiment?

The half-life of radium-226 is 1,590 years. How much of a 100 mg sample will be left in 500 years?

The half-life of magnesium-27 is 9.45 minutes. How much of a 10-mg sample will be left in 6 minutes?

The half-life of radioactive iodine is 60 days. How much of a 50-mg sample will be left in 40 days?

Solve for x.

$\text{log} 49 = 2 \text{log} x$

Solve for x.

$3 \text{log}_{6} x = \text{log}_{6} 64$

Solve for x.

$\text{log}_{3} \left(x^{2} + 3\right) = \text{log}_{3} 4 x$

Solve for x.

$\text{log}_{4} x + \text{log}_{4} x = 3$

Solve for x.

$\text{log}_{3} x + \text{log}_{3} \left(x + 6\right) = 3$

Solve for x.

$\text{log} x + \text{log} \left(x - 15\right) = 2$

Solve for x.

$\text{log} \left(x - 1\right) - \text{log} \left(x + 3\right) = \text{log} \frac{1}{x}$

Solve for x.

$\text{log}_{5} \left(x + 1\right) + \text{log}_{5} \left(x - 5\right) = \text{log}_{5} 7$

Solve for x.

$\text{log} \left(5 x + 1\right) = \text{log} \left(x + 3\right) + \text{log} 2$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$2^{x} = 74$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$4^{x} = 112$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$e^{x} = 8$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$\left(\frac{1}{3}\right)^{x} = 8$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$3 e^{x + 2} = 9$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$2 e^{3 x} = 32$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$\frac{1}{3} e^{x} = 2$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$e^{x - 1} + 4 = 12$

Solve each equation.

$6^{4 x - 17} = 216$

Solve each equation.

$\frac{e^{x^{2}}}{e^{x}} = e^{20}$

Solve each equation.

$\text{log}_{a} 81 = 4$

Solve each equation.

$\text{ln} x = 9$

Solve each equation.

$\text{log}_{4} \left(7 x + 15\right) = 3$

Solve each equation.

$\text{ln} e^{6 x} = 18$

Solve each equation.

$7 \text{log}_{3} x = \text{log}_{3} 128$

Solve each equation.

$\text{log}_{9} x + \text{log}_{9} \left(x - 4\right) = \text{log}_{9} 12$

Solve each equation.

$\text{log}_{6} \left(x + 1\right) - \text{log}_{6} \left(4 x + 10\right) = \text{log}_{6} \frac{1}{x}$

Solve for x, giving an exact answer as well as an approximation to three decimal places.

$\left(\frac{1}{2}\right)^{x} = 10$

Solve for x, giving an exact answer as well as an approximation to three decimal places.

$8 e^{x + 5} = 56$

Solve.

Alice invests $\$ 15,000$ at age 30 from the signing bonus of her new job. She hopes the investments will be worth $\$ 30,000$ when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Solve.

Simone invests $\$ 8,000$ in an account that compounds interest quarterly and earns $5 \% .$ How long will it take for his money to double?

Solve.

Researchers recorded that a certain bacteria population declined from 800,000 to 500,000 in 6 hours after the administration of medication. At this rate of decay, how many bacteria will there be in 24 hours?

Solve.

A bacteria doubles its original population in 24 hours $\left(A = 2 A_{0}\right) .$ How big will its population be in 72 hours?

Solve.

Radioactive technetium-99m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material form a 0.5 ml injection will be in the body in 24 hours?

What is the difference between the equation for exponential growth versus the equation for exponential decay?

For each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f \left(x\right) = 4 x$ and
$g \left(x\right) = x^{2} + 3 x$

Evaluate the composition.

For functions
$f \left(x\right) = 2 x^{3} + 5$ and
$g \left(x\right) = 3 x^{2} - 7 ,$ find
ⓐ $\left(f \circ g\right) \left(−1\right)$
ⓑ $\left(g \circ f\right) \left(−2\right)$
ⓒ $\left(g \circ g\right) \left(1\right)$

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

$\left\{\right. \left(−3 , 0\right) , \left(−2 , −2\right) , \left(−1 , 0\right) , \left(0 , 1\right) ,$
$\left(1 , 2\right) , \left(2 , 1\right) , \left(3 , −1\right) \left.\right\}$

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

$\left\{\left(−3 , 10\right) , \left(−2 , 5\right) , \left(−1 , 2\right) , \left(0 , 1\right)\right\}$

Verify that the functions are inverse functions.

$f \left(x\right) = 3 x + 7$ and
$g \left(x\right) = \frac{x - 7}{3}$

Find the inverse of each function.

$f \left(x\right) = 6 x - 11$

Find the inverse of each function.

$f \left(x\right) = \frac{1}{x + 5}$

Solve each equation.

$2^{x^{2}} = 16$

Solve each equation.

$5^{x^{2} + 2 x} = \frac{1}{5}$

Solve each equation.

$\frac{e^{x^{2}}}{e^{15}} = e^{2 x}$

Solve.

Sayed deposits $\$ 20,000$ in an investment account. What will be the value of his investment in 30 years if the investment is earning $7 \%$ per year and is compounded continuously?

Solve.

In the last five years the population of the United States has grown at a rate of $0.7 \%$ per year to about 318,900,000. If this rate continues, what will be the population in 5 more years?

Convert from exponential to logarithmic form.

$10^{−3} = \frac{1}{1,000}$

Convert from exponential to logarithmic form.

$e^{y} = 16$

Convert each logarithmic equation to exponential form.

$5 = \text{log} 100,000$

Solve for x.

$\text{log}_{x} 125 = 3$

Solve for x.

$\text{log}_{\frac{1}{2}} \frac{1}{16} = x$

Find the exact value of each logarithm without using a calculator.

$\text{log}_{8} 1$

Solve each logarithmic equation.

$\text{ln} x = −3$

Solve each logarithmic equation.

$\text{ln} e^{3 x} = 24$

Use Logarithmic Models in Applications

What is the decibel level of a train whistle with intensity $10^{−3}$ watts per square inch?

Use the properties of logarithms to evaluate.

ⓐ $5^{\text{log}_{5} 13}$ ⓑ $\text{log}_{3} 3^{−9}$

Use the properties of logarithms to evaluate.

ⓐ $e^{\text{ln} 8}$ ⓑ $\text{ln} e^{5}$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log} 10,000 m$

Use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

$\text{ln} \frac{e^{5}}{2}$

Use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

$\text{log}_{4} \sqrt[7]{z}$

Use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$

Use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log}_{6} \sqrt[3]{\frac{7 x^{2}}{6 y^{3} z^{5}}}$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$3 \text{log}_{3} x + 7 \text{log}_{3} y$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\frac{1}{4} \text{log} y - 2 \text{log} \left(y - 3\right)$

Rounding to three decimal places, approximate each logarithm.

$\text{log}_{\sqrt{3}} 16$

Solve for x.

$\text{log}_{2} x + \text{log}_{2} \left(x - 2\right) = 3$

Solve for x.

$\text{log}_{4} \left(x - 2\right) + \text{log}_{4} \left(x + 5\right) = \text{log}_{4} 8$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$2^{x} = 101$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$\left(\frac{1}{3}\right)^{x} = 7$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$e^{x - 4} + 8 = 23$

Use Exponential Models in Applications

Elise invests $\$ 4500$ in an account that compounds interest monthly and earns $6 \% .$ How long will it take for her money to double?

Use Exponential Models in Applications

Mouse populations can double in 8 months $\left(A = 2 A_{0}\right) .$ How long will it take for a mouse population to triple?

For the functions, $f \left(x\right) = 6 x + 1$ and $g \left(x\right) = 8 x - 3 ,$ find ⓐ $\left(f \circ g\right) \left(x\right) ,$ ⓑ $\left(g \circ f\right) \left(x\right) ,$ and ⓒ $\left(f \cdot g\right) \left(x\right) .$

Find the inverse of the function $f \left(x\right) = x^{5} - 9 .$

Solve the equation $2^{2 x - 4} = 64 .$

Megan invested $\$ 21,000$ in a savings account. If the interest rate is $5 \% ,$ how much will be in the account in 8 years by each method of compounding?
ⓐ compound quarterly
ⓑ compound monthly
ⓒ compound continuously.

Convert the equation from logarithmic equation to exponential form: $3 = \text{log}_{7} 343$

Evaluate $\text{log}_{11} 1 .$

What is the decibel level of a small fan with intensity $10^{−8}$ watts per square inch?

Use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

$\text{log}_{5} 25 a b$

Use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

$\text{log}_{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}$

Use the Properties of Logarithms to condense the logarithm, simplifying if possible.

$\frac{1}{6} \text{log} x - 3 \text{log} \left(x + 5\right)$

Use the Properties of Logarithms to condense the logarithm, simplifying if possible.

Solve for x:
$\text{log}_{7} \left(x + 2\right) + \text{log}_{7} \left(x - 3\right) = \text{log}_{7} 24$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

$5 e^{x - 4} = 40$

Solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

Researchers recorded that a certain bacteria population grew from 500 to 700 in 5 hours. At this rate of growth, how many bacteria will there be in 20 hours?