Question

solve the exponential equation. express the solution in terms of natural logarithms or common logarithms. then use a calculator to obtain a decimal approximation for the solution
10^x = 2.32

the solution set expressed in terms of logarithms is
(use a comma to separate answers as needed. simplify your answer. use integers or decimals for any numbers in the expression. use ln for natural logarithm and log for common logarithm)
now use a calculator to obtain a decimal approximation for the solution.
the solution set is
(use a comma to separate answers as needed. round to two decimal places as needed)

Explanation:

Step1: Take log of both sides

To solve \(10^x = 232\), take the common logarithm (log base 10) of both sides. By the logarithm power rule, \(\log(10^x)=x\log(10)\). Since \(\log(10) = 1\), we get \(x=\log(232)\) (or using natural logarithm: take \(\ln\) of both sides, \(\ln(10^x)=x\ln(10)\), so \(x = \frac{\ln(232)}{\ln(10)}\)).

Step2: Approximate the value

Using a calculator, \(\log(232)\approx2.3657\) (or \(\frac{\ln(232)}{\ln(10)}\approx\frac{5.4472}{2.3026}\approx2.3657\)). Rounding to two decimal places, we get \(2.37\).

Answer:

In terms of logarithms: \(x = \log(232)\) (or \(x=\frac{\ln(232)}{\ln(10)}\)); Decimal approximation: \(2.37\)