Animated Solution for Mathematics - Basic Mathematics: If $3^x=4^{x-1}$, then $x=$

We are tasked with solving the exponential equation $3^x=4^{x-1}$.

At first glance, this appears to be a standard exponential problem. However, the true challenge lies in manipulating the solution into various equivalent logarithmic forms.

To solve for $x$, we must bring the variable down from the exponent. We apply the logarithm to both sides of the equation:

Using the power rule of logarithms, $\log (a^b)=b\log a$, we transform the equation into:

The variable $x$ is now accessible for algebraic manipulation.

Next, we expand the right side of the equation to isolate the terms containing $x$:

Rearranging the terms to group all $x$ variables on one side, we obtain:

Factoring out $x$ from the right side gives:

Dividing both sides, we arrive at the base solution:

To match various potential answer formats, we utilize the base change formula, $\frac{\log a}{\log b}={\log }_{b}a$.

To derive the form involving ${\log }_{4}3$, we divide the numerator and denominator by $\log 4$:

To derive the form involving ${\log }_{2}3$, we substitute $\log 4=2\log 2$:

Dividing the numerator and denominator by $\log 2$ yields:

Finally, to express the result in terms of ${\log }_{3}2$, we divide the original expression by $\log 3$:

In a JEE Advanced context, finding a single form of the answer is often insufficient. You must be fluent in the language of logarithms to transform your result into the specific structure demanded by the options.

Never stop at the first answer you find. Always analyze the provided options and use your algebraic toolkit to bridge the gap between your result and the examiner's expected format.