How to find an unknown exponent?

Understand the Problem

The question is asking for a method or approach to determine an unknown exponent in a mathematical expression or equation. This typically involves using logarithms or algebraic manipulation depending on the context of the problem.

Answer

$$ x = \frac{\ln(b)}{\ln(a)} $$

Steps to Solve

Identify the equation with the unknown exponent
Look for the equation that contains the unknown exponent, which we'll denote as $x$. For instance, if the equation is of the form $a^x = b$, where $a$ and $b$ are known constants.
Take the logarithm on both sides
Use logarithms to eliminate the exponent. You can use any logarithmic base, but the natural logarithm ($\ln$) is commonly used. The equation becomes:
$$ \ln(a^x) = \ln(b) $$
Apply the power rule of logarithms
Use the power rule of logarithms which states that $\ln(a^x) = x \ln(a)$. This transforms the equation into:
$$ x \ln(a) = \ln(b) $$
Isolate the unknown exponent
Solve for $x$ by dividing both sides of the equation by $\ln(a)$:
$$ x = \frac{\ln(b)}{\ln(a)} $$
Calculate the numerical value
Substitute the values of $a$ and $b$ into the equation to compute $x$.

The unknown exponent $x$ can be found using the formula:
$$ x = \frac{\ln(b)}{\ln(a)} $$

More Information

This method is based on the properties of logarithms and is very useful in various applications, especially in solving exponential equations in algebra. Logarithmic identities simplify solving for unknown variables when they are in the exponent.

Tips

Forgetting to take the logarithm on both sides, which can lead to an incorrect conclusion.
Not applying the power rule correctly when dealing with logarithms, which can result in an incorrect expression.
Confusing the base of the logarithm, which can affect the final answer if a different base is chosen.

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