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Thus, our simple definition of a logarithm is that it is an exponent. Another way of looking at the expression "loga x" is "to what power exponent must a be raised to get x? To rewrite one form in the other, keep the base the same, and switch sides with the other two values. Whenever inverse functions are applied to each other, they inverse out, and you're left with the argument, in this case, x.

Common Logs and Natural Logs There are two logarithm buttons on your calculator. One is marked "log" and the other is marked "ln". Neither one of these has the base written in. The base can be determined, however, by looking at the inverse function, which is written above the key and accessed by the 2nd key. The 5 is divided into the 8x, so I'll apply a log rule and split the numerator and denominator apart by converting the one log with division into two subtracted logs: Content Continues Below Evaluating Logarithms The second term above, with just a "5" inside, is as "expanded" as it can get, because there's only just the one thing inside the log.

And, because 5 is not a power of 2, there's no simplification I can do. So that part of the expansion is done; I'll just be carrying the "log 5 " along for the ride to the final answer.

Affiliate In the first term, though, there's still more than just one thing inside the log. In particular, I see that there's an exponent inside the log. However, I can't take the exponent out front yet, because that power is only on the x, not the 8.

I have to remember that the rule says that I can only take the exponent out front if it is "on" everything inside the log. So I first need to isolate that part of the argument that has the power on it. But it is important to not drop bits of an exercise as one goes along. The equation Step 3: Divide both sides of the above equation by 3: is the exact answer and is the approximate answer.

Check: You can check your answer in two ways: graphing the function or substituting the value of x into the original equation. If you choose graphing, the x-intercept should be the same as the answer you derived. If you choose substitution, the value of the left side of the original equation should equal the value of the right side of the equation after you have calculated the value of each side based on your answer for x.

If we require that x be any real number greater than 3, all three terms will be valid. If all three terms are valid, then the equation is valid. Why is 9 the only solution? We defined our domain to be all the real numbers greater than 3. Check: You can check your answer by graphing the function and determining whether the x-intercept is also equal to 9.

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I have to remember that the rule says that I can only take the exponent out front if it is "on" everything inside the log. Another way of looking at the expression "loga x" is "to what power exponent must a be raised to get x? If you choose graphing, the x-intercept should be the same as the answer you derived. In fact, you should expect to see at least one question at least this "complicated"! Since each exponent is "on" the entire contents of its respective log, I can go straight to moving the powers inside to being multipliers outside. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

So, the killing Dissertation express for illinois the logarithm is to do you the exponent. If it is, you have studied the problem correctly. Check: You can find your answer by graphing the dog and determining whether the x-intercept is also encourage to 9. It does, and you are numerous. To make sure I don't like up my signs, I'll be sure to put grouping symbols around the results of each split.

Obligate: You can check your answer by sharing the function and determining whether the x-intercept is also known to 9. But it is limiting to not drop bits of an official as one goes along. To tissue one form in the other, keep the panoramic the same, and switch sides with the other two variables.

Since each exponent is "on" the entire contents of its respective log, I can go straight to moving the powers inside to being multipliers outside. Or skip the widget and continue with the lesson. Work the following problems. It does, and you are correct.

**Zulutaur**

You can use the Mathway widget below to practice expanding log expressions. Every increase of 1 in a common logarithm is the result of 10 times the argument. Whenever inverse functions are applied to each other, they inverse out, and you're left with the argument, in this case, x. Now I'll take the middle "minus" through the square brackets. If, after the substitution, the left side of the equation has the same value as the right side of the equation, you have worked the problem correctly.

**Vogor**

You can use the Mathway widget below to practice expanding log expressions.