Solve the exponential equation. Express the solution in terms of natural logarithms. e^x=9.2 ...

Question:

Solve the exponential equation. Express the solution in terms of natural logarithms. {eq}e^x=9.2 {/eq}

What is the solution in terms of natural logarithms? Type a decimal rounded to 3 decimal places.

Solving an Exponential Equation

When we try to solve an equation where our variable is within an exponent, we need to first simplify it in a way that causes our variable to not be within an exponent. We can do this by applying logarithms to each side of an equation. This is because the logarithm is the inverse of an exponential function. If we use a logarithm that has the same base as the exponential expression, this leaves only the variable or expression inside of the exponent.

Since this is an exponential expression, we need to use logarithms to solve for x. Since the left side of this equation contains only this exponential piece without any coefficients, we don't have any preliminary simplification to do beforehand, and we can jump right to taking the logarithm of both sides. The exponential expression has a base of e, so we need to use a logarithm that also has a base of e, which is the natural logarithm. This will eliminate the exponential part of the left side, leaving only our variable. We can also type this into our calculator to get a decimal approximation, as this solution is likely to be an irrational number.

{eq}e^x=9.2\\ \ln e^x = \ln 9.2\\ x = \ln 9.2 \approx 2.2192035 {/eq}