Solve each equation. Give solutions in exact form. See Examples 5...

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (log_2 x) = 1

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're gonna be looking at this question that's asking us to solve the logarithmic equation for X and provide an exact solution where our equation is log base five of log base five of X is equal to one. Our answer choices show up as A is 25. B is one C is 3125 and D is five. So let's work through this little by little step by step first by writing down the information that they give us, which is our equation log base five of log base five again of X is equal to one. Now, although it looks a little more complicated at the end of the day, this is just another logarithmic equation. It's a simple algorithm equation that can be turned into an exponential equation by simply recalling the general formula on how to do that. So to recall, let's just give a general equation log base A of B is equal to C. Now, that algorithm can be rewritten as an exponential equation as A to the exponent C is equal to B and this is a quick recall on how to switch between algorithms and exponent equations which we, we can apply to the general equation that we have here where the log base five of X would function as the B, the initial base five would function as the A and the number one would function as the C. So by using that general formula and the recall formula, we can rewrite our initial equation as five to the exponent one is equal to log base five of X. And now we know that five to the exponent one is just five. And once again, we have another logarithm equation that can be easily turned into an exponential equation. So we look at our, we call formula once again and we see that the number five in, in the answer here would function as the C, the base five of the log would function as the A and the X variable would function as the B. So once again, using our recall formula, we can write this as five to the fifth power is equal to X. And we know that five to the exponent five will give us 3125. And that is equal to X. And that would be our answer to this question or letter C. I hope that was helpful. And until next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (log_2 x) = 1

Key Concepts

Logarithmic Functions

Change of Base Formula

Nested Logarithms

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