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 = 311 \\) the solution set expressed in terms of logarithms is \\( \boxed{} \\). (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: Identify the exponential equation

The equation is \( 10^x = 311 \). To solve for \( x \), we use the property of logarithms that if \( a^b = c \), then \( b=\log_a c \). For base 10 (common logarithm), we can rewrite the equation using the common logarithm (\(\log\)) or natural logarithm (\(\ln\)).

Step2: Solve for \( x \) using logarithms

Taking the common logarithm (base 10) of both sides:
\(\log(10^x) = \log(311)\)
Using the logarithm power rule \( \log(a^b)=b\log(a) \), we get:
\( x\log(10) = \log(311) \)
Since \( \log(10) = 1 \), this simplifies to:
\( x = \log(311) \)

If using natural logarithm (\(\ln\)):
Take \( \ln \) of both sides: \( \ln(10^x) = \ln(311) \)
Using the power rule: \( x\ln(10) = \ln(311) \)
Then \( x = \frac{\ln(311)}{\ln(10)} \) (by the change - of - base formula, this is equivalent to \( \log(311) \)).

Step3: Calculate the numerical value

Using a calculator, \( \log(311)\approx2.49277 \) (or \( \frac{\ln(311)}{\ln(10)}\approx\frac{5.737}{2.3026}\approx2.49277 \)).

Answer:

In terms of logarithms, \( x = \log(311) \) (or \( x=\frac{\ln(311)}{\ln(10)} \)) and the decimal approximation is approximately \( 2.49 \) (or more precisely \( 2.4928 \)).