Answer – The exponent x in the given equation can be solved for using logarithms, and its value is found to be 2.0704.

Explanation:

Let’s begin by reviewing some fundamental information about logarithms. 

A logarithm is another way of expressing an exponent. It is defined as a power to which a number must be raised to get another number.

To understand this better, let us consider the example “3 raised to the power 4 equals 81.” This can be expressed in the form of the below exponential equation:

$3^4 = 81$

The same equation can also be expressed in the logarithmic form by answering the question 3 raised to what power equals 81. The answer, which is the power 4, takes the right side of the logarithmic equation, while the left side is simply log 81 with the base 3. So the logarithmic equation will be:

${\log }_{3}81 = 4$

This is read as “log base 3 of 81 equals 4.”

It is also evident that both the exponential and logarithmic equations have:

• The same base – 3
• The same exponent – 4
• The same argument – 81

Now, let us consider the given equation $30 = 200 {(0.4)}^x$.

Since there is an additional number 200 next to the number raised to x, we need to simplify the equation further before solving for x. We can do this by dividing both sides by 200.

So the equation becomes $\frac{30}{200} = {(0.4)}^x$.

$\Rightarrow 0.15 = {(0.4)}^x$

Now applying log on both sides, we get:

$\log 0.15 = x \log 0.4$ (An exponent in a logarithmic expression is always moved before the log and multiplied with it.)

We now cross-multiply to get the value of x:

$x = \frac{\log 0.15}{\log 0.4} = \frac{–0.8239}{–0.3979} = 2.0704$

Thus, $x = 2.0704$.

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