Using the property that \(\ln(e^y) = y\), this simplifies to:
\[
\log_x(2) = \ln(5)
\]
Rewriting in Exponential Form
To find \(x\), rewrite the expression in exponential form:
\[
x^{\ln(5)} = 2
\]
Take the inverse power with respect to \(\ln(5)\):
\[
x = 2^{1/\ln(5)}
\]
Checking the Solution
To verify the solution, substitute the derived value for \(x\) into the original expression:
\[
e^{\log_{2^{1/\ln(5)}}(2)} = e^{\ln(5)}
\]
Since \(e^{\ln(5)} = 5\), the solution checks out, confirming that \(x = 2^{1/\ln(5)}\) solves the original equation.
Start with an Exponential Expression: The initial equation contains an exponential expression where the exponent involves a logarithmic function.
Use Natural Logarithms: To isolate the logarithmic term, natural logarithms are applied to both sides. This step simplifies the expression by using the property that the natural logarithm of an exponential expression is the exponent itself.
Rewrite in Exponential Form: The logarithmic expression is rewritten in its equivalent exponential form to isolate and solve for a specific variable.
Find the Solution: By manipulating the rewritten expression, the value of the unknown variable is obtained.
Verify the Solution: To ensure the solution is correct, the derived value is substituted back into the original expression. This step checks if the expression yields the expected result.
Overall, this sequence of operations demonstrates how to use mathematical techniques to isolate and solve for a variable in a complex expression.
