Example 10.24: Solving $\log_a 49 = 2$ and $\ln x = 3$

Try It 10.47: Solving $\log_a 121 = 2$ and $\ln x = 7$

Try It 10.48: Solving $\log_a 64 = 3$ and $\ln x = 9$

Example 10.25: Solving $\log_2(3x - 5) = 4$ and $\ln e^{2x} = 4$

Try It 10.49: Solving $\log_2(5x - 1) = 6$ and $\ln e^{3x} = 6$

Try It 10.50: Solving $\log_3(4x + 3) = 3$ and $\ln e^{4x} = 4$

Example 10.20: Evaluating Logarithmic Equations

Try It 10.39: Solving $\log_x 64 = 2$, $\log_5 x = 3$, and $\log_{\frac{1}{2}} \frac{1}{4} = x$

Try It 10.40: Solving $\log_x 81 = 2$, $\log_3 x = 5$, and $\log_{\frac{1}{3}} \frac{1}{27} = x$

Extraneous Solution to a Logarithmic Equation

Example 10.39: Solving $\log_3 x + \log_3(x-8) = 2$

Try It 10.77 and 10.78: Solving $\log_2 x + \log_2(x-2) = 3$ and $\log_2 x + \log_2(x-6) = 4$

As a data technician at a logistics company, you are completing a certification on mathematical models for shipment scaling. A module asks you to solve the logarithmic equation $\log_5(x) = 3$ to find the optimal cargo volume. According to the foundational rule for solving logarithmic equations, what is the correct equivalent exponential form of this equation?

A laboratory technician is following a standard operating procedure (SOP) to calculate the decay of a radioactive isotope using a logarithmic model. To ensure the results are mathematically sound, the technician must solve the model's equation correctly. Arrange the following steps in the correct order to solve a basic logarithmic equation by converting it to exponential form.

A supply chain analyst uses a logarithmic growth model to predict inventory turnover. To find the optimal stock level $x$, the analyst needs to solve the equation $\log_{6}(x) = 2$. True or False: The analyst can correctly rewrite this equation in the exponential form ${}6^2 = x$ to begin solving for $x$.

Domain Constraints in Logarithmic Equations

As a logistics data analyst, you are auditing growth models used to predict delivery volumes. To ensure the calculations are correct, you must match each logarithmic model with its equivalent exponential form. Match the logarithmic equations on the left with their corresponding exponential conversions on the right.

As a member of a financial planning team at a local credit union, you are drafting an instructional guide on calculating savings doubling times. You want to remind learners that solving logarithmic equations can sometimes produce values that do not satisfy the original equation because they result in a negative argument. These invalid values are called _________ solutions.

Troubleshooting Logarithmic Resistance Models