Solving exponential and logarithmic equation. An example of an exponential equation is:

Solve for x:

5^x=125

This one is easy ....

5^x=125
5^x = 5^3 , so x = 3

fortunately 125 was a power of 5, unfortunately this will be the case rarely.

To solve the exponential equation 5^x = 125, we need to find the value of x that makes this equation true.

One approach to solving this equation is by using the logarithmic function. The logarithmic function is the inverse of an exponential function, and it can help us isolate the exponent.

In this case, we can take the logarithm of both sides of the equation. The most commonly used logarithm is the natural logarithm (ln), but we can also use logarithms with any other base, such as logarithm base 10 (log).

So, let's take the logarithm of both sides using the natural logarithm (ln) since it is a commonly used logarithm:

ln(5^x) = ln(125)

Now, we can use the logarithmic property that states that the logarithm of a number raised to an exponent can be written as the exponent multiplied by the logarithm of the base:

x * ln(5) = ln(125)

To find the value of x, we need to isolate it. In this case, x is being multiplied by ln(5), so we can divide both sides of the equation by ln(5):

x = ln(125) / ln(5)

Now, we can use a calculator to evaluate this expression:

x ≈ 3

Hence, the solution to the exponential equation 5^x = 125 is approximately x = 3.