Question 269804
Your equation makes no sense. The parentheses are not even balanced. So I cannot help you with it.<br>
But here are some general tips:<ol><li>Isolate the base and its exponent with a variable.</li><li>Find the logarithm of each side of the equation. Generally the base you use does not matter. If you want a decimal approximation of the answer, use a base for the logarithms which your calculator "knows" (like base 10 or base e (ln)). Sometimes it is helpful to choose the base of the logarithm to match the base on which there is an exponent with the variable.</li></li>After steps #1 and #2, you should have an equation where one side is the log of a number with an exponent that has the variable for which you're trying to solve. On this logarithm, use the property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, to move the exponent in the argument out in front. This is how you get a variable out of an exponent.</li><li>Solve the resulting equation for the variable.</ol>
Here's an example:
{{{8 = 3*4^(x-3) - 7}}}
1. Isolate the base and its exponent. Add 7 to both sides:
{{{15 = 3*4^(x-3)}}}
Divide both sides of the equation by 3:
{{{5 = 4^(x-3)}}}
2. Find the log of each side. If we want a simple, exact answer, use base 4 logarithms. If we want a decimal approximation, use base 10 or base e logarithms. I will base 4. (At the end I will use base 10 logarithms so you can see that, too.)
{{{log(4, (5)) = log(4, (4^(x-3)))}}}
3. Use the property of logarithms to move the exponent out in front:
{{{log(4, (5)) = (x-3)log(4, (4))}}}
Since {{{log(4, (4)) = 1}}} (which is why I chose base 4 logarithms) this becomes:
{{{log(4, (5)) = x-3}}}
4. Solve the equation. All we need to do is add 3 to each side:
{{{log(4, (5)) + 3 = x}}}
This is an exact answer to the example problem.<br>
At step #2, if we use base 10 logarithms instead, we get:
{{{log((5)) = log((4^(x-3)))}}}
Then, using the property of logarithms we get:
{{{log((5)) = (x-3)log((4))}}}
And to solve for x we divide both sides by log(4):
{{{log((5))/log((4)) = x-3}}}
and add 3 to each side:
{{{log((5))/log((4)) + 3 = x}}}
This is also an exact answer to the example equation. And if we use our calculators we can find a decimal approximation of the answer:
{{{0.6989700043360188/log((4)) + 3 = x}}}
{{{0.6989700043360188/0.6020599913279624 + 3 = x}}}
{{{1.1609640474436812 + 3 = x}}}
{{{4.1609640474436812 = x}}}