6.6 Exponential and Logarithmic Equations

Solve: $3^{2 x - 5} = 27 .$

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

Solve: $\log_{2} \left(3 x - 5\right) = 4$

Solve $\log_{4} \left(x + 6\right) - \log_{4} \left(2 x + 5\right) = - \log_{4} x$

Solve $2^{x - 1} = 2^{2 x - 4} .$

Solve $5^{2 x} = 5^{3 x + 2} .$

Solve $8^{x + 2} = 16^{x + 1} .$

Solve $5^{2 x} = 25^{3 x + 2} .$

Solve $2^{5 x} = \sqrt{2} .$

Solve $5^{x} = \sqrt{5} .$

Solve $3^{x + 1} = −2.$

Solve $2^{x} = −100.$

Solve $5^{x + 2} = 4^{x} .$

Solve $2^{x} = 3^{x + 1} .$

Solve $100 = 20 e^{2 t} .$

Solve $3 e^{0.5 t} = 11.$

Solve $4 e^{2 x} + 5 = 12.$

Solve $3 + e^{2 t} = 7 e^{2 t} .$

Solve $e^{2 x} - e^{x} = 56.$

Solve $e^{2 x} = e^{x} + 2.$

Solve $2 \ln x + 3 = 7.$

Solve $6 + \ln x = 10.$

Solve $2 \ln \left(6 x\right) = 7.$

Solve $2 \ln \left(x + 1\right) = 10.$

Use a graphing calculator to estimate the approximate solution to the logarithmic equation $2^{x} = 1000$ to 2 decimal places.

Solve $\ln \left(x^{2}\right) = \ln \left(2 x + 3\right) .$

Solve $\ln \left(x^{2}\right) = \ln 1.$

How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?

How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?

How can an exponential equation be solved?

When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

Use like bases to solve the exponential equation.

$64 \cdot 4^{3 x} = 16$

Use like bases to solve the exponential equation.

$2^{- 3 n} \cdot \frac{1}{4} = 2^{n + 2}$

Use like bases to solve the exponential equation.

$\frac{36^{3 b}}{36^{2 b}} = 216^{2 - b}$

Use logarithms to solve.

$9^{x - 10} = 1$

Use logarithms to solve.

$e^{r + 10} - 10 = −42$

Use logarithms to solve.

$- 8 \cdot 10^{p + 7} - 7 = −24$

Use logarithms to solve.

$e^{- 3 k} + 6 = 44$

Use logarithms to solve.

$- 6 e^{9 x + 8} + 2 = −74$

Use logarithms to solve.

$e^{2 x} - e^{x} - 132 = 0$

Use logarithms to solve.

$10 e^{8 x + 3} + 2 = 8$

Use logarithms to solve.

$8 e^{- 5 x - 2} - 4 = −90$

Use logarithms to solve.

$e^{2 x} - e^{x} - 6 = 0$

Use the definition of a logarithm to rewrite the equation as an exponential equation.

$\log \left(\frac{1}{100}\right) = −2$

Use the definition of a logarithm to solve the equation.

$5 \log_{7} n = 10$

Use the definition of a logarithm to solve the equation.

$4 + \log_{2} \left(9 k\right) = 2$

Use the definition of a logarithm to solve the equation.

$10 - 4 \ln \left(9 - 8 x\right) = 6$

Use the one-to-one property of logarithms to solve.

$\log_{13} \left(5 n - 2\right) = \log_{13} \left(8 - 5 n\right)$

Use the one-to-one property of logarithms to solve.

$\ln \left(- 3 x\right) = \ln \left(x^{2} - 6 x\right)$

Use the one-to-one property of logarithms to solve.

$\ln \left(x - 2\right) - \ln \left(x\right) = \ln \left(54\right)$

Use the one-to-one property of logarithms to solve.

$\ln \left(x^{2} - 10\right) + \ln \left(9\right) = \ln \left(10\right)$

Solve each equation for $x .$

$\ln \left(x\right) + \ln \left(x - 3\right) = \ln \left(7 x\right)$

Solve each equation for $x .$

$\ln \left(7\right) + \ln \left(2 - 4 x^{2}\right) = \ln \left(14\right)$

Solve each equation for $x .$

$\ln \left(3\right) - \ln \left(3 - 3 x\right) = \ln \left(4\right)$

Solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$e^{5 x} = 17$ using the natural log

Solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

$3^{4 x - 5} = 38$ using the common log

Use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

$7 e^{3 x - 5} + 7.9 = 47$

Use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

$\log \left(- 0.7 x - 9\right) = 1 + 5 \log \left(5\right)$

Use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

The magnitude M of an earthquake is represented by the equation $M = \frac{2}{3} \log \left(\frac{E}{E_{0}}\right)$ where $E$ is the amount of energy released by the earthquake in joules and $E_{0} = 10^{4.4}$ is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing $1.4 \cdot 10^{13}$…

Recall the formula for continually compounding interest, $y = A e^{k t} .$ Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.

Newton’s Law of Cooling states that the temperature $T$ of an object at any time t can be described by the equation $T = T_{s} + \left(T_{0} - T_{s}\right) e^{- k t} ,$ where $T_{s}$ is the temperature of the surrounding environment, $T_{0}$ is the initial temperature of the object, and $k$ is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.