Answer:
x = -3
Explanation:
(5^3)^2 · 5^(x+4) = 5^7
First of all, we simplify (5^3)^2. When you have an exponent in parentheses that is raised to another exponent that is outside the parentheses, you multiply the exponents.
(5^3)^2 = 5^(3·2) = 5^6
We cannot simplify 5^(x+4) or 5^7 any further, so our equation is now:
5^6 · 5^(x+4) = 5^7
We can divide 5^6 from both sides to get:
5^(x+4) = (5^7)/(5^6)
When exponents of like terms are divided we can subtract the exponents.
5^(x+4) = 5^(7-6)
5^(x+4) = 5^1
This last part is a bit trickier. When exponents of like terms are multiplied we add the exponents. We can use this knowledge to determine that 5^(x+4) is made up of 5^x · 5^4. Now we have:
5^x · 5^4 = 5^1
We can divide both sides by 5^4.
5^x = (5^1)/(5^4)
We simplify the right side by the same way we did earlier when we divided exponents:
5^x = 5^(1-4)
5^x = 5^-3
We can see that x = -3, but using logarithms, we can finish isolating x. Taking the log base 5 of both sides, we get:
x = log₅(5^-3)
If you don't already know, log₅(5^-3) means what exponent do you raise 5 to in order to get 5^-3. After stating it like this, we can clearly see that log₅(5^-3) equals -3.
So x = -3
Another way we could have solved the logarithm is by using one of the laws of exponents. In this case, we would use logₐ(x^y) = y(logₐm). This would give us:
x = -3(log₅5)
x = -3(1)
x = -3
Either way works.