Question

These are my last points. I just wanted to give them away so the first one to answer correctly get the last points.

Answer 1

Answer:

x = -3

Step-by-step 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.

Answer 2

Answer:

x=-3

Step-by-step explanation:

\left(5^3\right)^2\cdot 5^{x+4}=5^7

Apply\:exponent\:rule

x+10=7

x=-3