Question

solve the exponential equation. express the solution in terms of natural logarithms or common logarithms. then obtain a decimal approximation for the solution.
$10^x = 2.05$
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: Take common logarithm on both sides

To solve the exponential equation \(10^{x}=2.05\), we can take the common logarithm (base - 10) of both sides. Recall that if \(y = a^{x}\), then \(\log(y)=\log(a^{x})\) and by the power rule of logarithms \(\log(a^{x})=x\log(a)\).

Taking \(\log\) (common logarithm) on both sides of \(10^{x}=2.05\), we get:
\(\log(10^{x})=\log(2.05)\)

Step2: Apply power rule of logarithms

Using the power rule of logarithms \(\log(a^{b}) = b\log(a)\), for the left - hand side \(\log(10^{x})\), we have \(x\log(10)\). Since \(\log(10) = 1\) (because \(\log_{10}(10)=1\)), the equation becomes:
\(x\times1=\log(2.05)\)
So, \(x = \log(2.05)\)

If we want to use natural logarithm, we can use the change - of - base formula. Recall that \(\log_{a}(b)=\frac{\ln(b)}{\ln(a)}\). So, \(\log(2.05)=\frac{\ln(2.05)}{\ln(10)}\)

Step3: Calculate the decimal approximation

We know that \(\log(2.05)\approx0.3117\) (using a calculator, since \(\log(2)=0.3010\) and \(\log(2.05)\) is a bit more than \(\log(2)\)) or using the natural logarithm approach: \(\ln(2.05)\approx0.717\) and \(\ln(10)\approx2.3026\), so \(\frac{\ln(2.05)}{\ln(10)}\approx\frac{0.717}{2.3026}\approx0.311\)

Answer:

In terms of logarithms, the solution is \(x = \log(2.05)\) (or \(x=\frac{\ln(2.05)}{\ln(10)}\)) and the decimal approximation is \(x\approx0.312\) (rounded to three decimal places)