SOLUTION: Solve the exponential equation. Leave the answer in exact form. Do not approximate. {{{7^4x

SOLUTION: Solve the exponential equation. Leave the answer in exact form. Do not approximate. {{{7^4x - 5 = 10}}}

Question 267188: Solve the exponential equation. Leave the answer in exact form. Do not approximate.

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
I am assuming that the equation is

If not then a) It is not a problem for a logarithm; and b) repost your question more clearly.

With a variable in the exponent, we will use logarithms to solve it. But before we start with the logarithms, we should isolate the term with the variable. So we will start by adding 5 to each side:

Since 15 is not a known power of 7, we will use logarithms to finish. We can use a logarithm of any base to solve this. Normally, if you want a decimal approximation for the answer, then you should choose a logarithm your calculator "knows" (like base 10 or base e (ln)). However, since your problem states that the answer should be exact we do not want decimal approximations of the answer. So I will choose base 7 for the logarithm because it will make the answer simpler. (Later, I will also do the problem using base 10 logarithms.)
The next step, then, is to find the base 7 logarithm of each side:

On the left side we can use a property of logarithms, , to move the exponent out in front of the logarithm. (Using this property to get the variable out of the exponent is the reason we use logarithms on these problems!). After using the property we have:

Since we can simplify the left side further. (This is why I chose base 7 logarithms.)

And last of all we divide both sides by 4:

This is an exact answer to your problem.

Here's a solution using base 10 logarithms. We still find the logarithm of each side:

And use the property to move the exponent out in front:

Now we divide both sides by . If this confuses you, divide once by 4 and then divide a second time by . You should end up with the same thing as:

This is another exact answer for this problem. It looks a lot different from our earlier solution, , but they are equal.
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