Solving Exponential Equations Using Logarithms

Example 10.41: Solving $5^x = 11$

Try It 10.81 and 10.82: Solving $7^x = 43$ and $8^x = 98$

Example 10.42: Solving $3e^{x+2} = 24$

Try It 10.83 and 10.84: Solving $2e^{x-2} = 18$ and $5e^{2x} = 25$

Solving an Exponential Equation by Rewriting with a Common Base

In professional fields such as finance and technology, analysts use different mathematical models to track growth and change. Which of the following represents an exponential equation?

In a professional financial model, an analyst uses the equation ${}500(1.04)^t = 1000$ to predict how many years ($t$) it will take for an investment to double. This is classified as an exponential equation because the variable $t$ is located in the exponent of the expression.

Structural Characteristics of Exponential Equations

In a professional financial report, an analyst uses the formula $V = 1000(1.05)^t$ to predict the future value of an investment over time $t$. Because the variable representing time is located in the exponent of the expression, this is specifically classified as a(n) ____ equation.

Auditing Business Growth Models

As a business analyst, you regularly work with mathematical equations to model company performance, growth, and depreciation. Match each equation used in business forecasting to its correct algebraic classification.

As a junior business analyst auditing growth models, you need to verify whether a newly proposed formula is classified as an 'exponential equation.' Arrange the steps of the structural analysis in the correct logical sequence, starting with analyzing the variable and ending with confirming the equation structure.