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 = 3.58
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.)

Explanation:

Step1: Take common logarithm on both sides

To solve the exponential equation \(10^{x}=3.58\), we take the common logarithm (base - 10) of both sides. By the property of logarithms, if \(a^{b}=c\), then \(\log(a^{b})=\log(c)\), and using the power rule of logarithms \(\log(a^{b}) = b\log(a)\). So, taking \(\log_{10}\) (or simply \(\log\)) of both sides of \(10^{x}=3.58\), we get:
\(\log(10^{x})=\log(3.58)\)
Since \(\log(10^{x})=x\log(10)\) and \(\log(10) = 1\), this simplifies to:
\(x=\log(3.58)\)

We could also use natural logarithm. If we take the natural logarithm (\(\ln\)) of both sides:
\(\ln(10^{x})=\ln(3.58)\)
Using the power rule \(\ln(a^{b})=b\ln(a)\), we have \(x\ln(10)=\ln(3.58)\), so \(x = \frac{\ln(3.58)}{\ln(10)}\) (by the change - of - base formula \(\log_{a}(b)=\frac{\ln(b)}{\ln(a)}\))

Step2: Calculate the numerical value

Now, we use a calculator to find the value of \(\log(3.58)\) (or \(\frac{\ln(3.58)}{\ln(10)}\)).
Using a calculator, \(\log(3.58)\approx0.5539\) (and \(\frac{\ln(3.58)}{\ln(10)}\approx\frac{1.275}{2.3026}\approx0.5539\))

Answer:

In terms of common logarithms, \(x = \log(3.58)\approx0.55\) (or in terms of natural logarithms \(x=\frac{\ln(3.58)}{\ln(10)}\approx0.55\))