**
Let a, b, and x be positive real numbers such that neither a nor b equals 1.
Then can be converted to the base b by the formula . Let's verify this with the following example.**

**Example 1:** Find .

**Answer 1:** -36.158608

**Solution:** Convert to the base 10.

**Check:** Let's check the answer. If
equals 125, we have worked the problem correctly. = 124.999443. Close enough. The reason the check
does not come out exactly is that we rounded the log125, the , and the quotient of the two logs. If you use continuous calculations
on your calculator, the answer will be closer to 125.

**Solution 2:** You can also work this problem using natural logarithms. Convert
to the base e.

**Check:** Let's check the answer. If
equals 125, we have worked the problem correctly. . Close enough. The reason the check
does not come out exactly is that we rounded the ln125, the , and the quotient of the two logs. If you use continuous calculations on
your calculator, the answer will be closer to 125.

**If you would like to review another example of changing the base of
a logarithm, click on Example.**

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