You can put this solution on YOUR website!
So solve this we must extract the x from the logarithm. To do this we can just rewrite the equation in exponential form. In general is equivalent to . Using this pattern on your equation we get:
Now we can solve for x. Perhaps the fastest solution is to find a way to turn the exponent of -3 in to an exponent of 1. This can be done by raising each side to the reciprocal of -3 power. The reciprocal of -3 is -1/3:
On the left side, where we have a power of a power, the rule is to multiply exponents. And when you multiply reciprocals you always get 1! (This rule and the fact that the product of reciprocals is always a 1 is the reason raising both sides to the reciprocal power "converts" the previous expoennt of -3 into a 1.)
All that is left is to simplify the right side. If you have trouble with negative and/or fractional exponents I find it can help if you factor the exponent in a special way:
If the exponent is negative, factor out -1.
If the exponent is a fraction and its numerator is not 1, then factor out the numerator.
Our exponent is negative we we will factor out -1:
The exponent has a fraction as a factor but its numerator is 1 so we are finished factoring the exponent. With the exponent factored each factor tells us an operation to perform.
The factor of -1 in the exponent tells us to find a reciprocal. The factor of 1/3 in the exponent tells us that we must find a cube root. And it makes no difference which operation we do first! So let's do the operations in the order that seems easiest. So we either find the reciprocal of 125 first or find the cube root of 125 first. The reciprocal of 125 will be a fraction (1/125). This is not hard to do but then we will have a fraction to work with for the cube root part. The cube root of 125 is not especially hard to figure out. It doesn't take long to find that . So the cube root is just 5. I think the cube root is easier to start with so we'll start with the cube root:
(Note how the factor gets removed once the operation has been done.) And we can finish with the reciprocal:
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