Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^0.3x=813

Verified step by step guidance

1. Start with the given exponential equation: \$7^{0.3x} = 813$.
2. To solve for \(x\), take the natural logarithm (or common logarithm) of both sides to utilize the property that \(\ln(a^b) = b \ln(a)\): \(\ln(7^{0.3x}) = \ln(813)\).
3. Apply the logarithm power rule to bring down the exponent: \(0.3x \cdot \ln(7) = \ln(813)\).
4. Isolate \(x\) by dividing both sides by \(0.3 \ln(7)\): \(x = \frac{\ln(813)}{0.3 \cdot \ln(7)}\).
5. Use a calculator to find the decimal values of the logarithms and compute the quotient to get the approximate value of \(x\) rounded to two decimal places.

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves rewriting the equation to isolate the exponential expression and then applying logarithms to both sides to solve for the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials. They allow us to solve equations where the variable is an exponent by converting the exponential form into a logarithmic form. Common logarithms (base 10) and natural logarithms (base e) are frequently used.

Using Calculators for Approximation

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to the desired decimal places, ensuring practical and usable answers.