Solve the exponential equation. 4 x + 7 = 16 4^{x+7}=16 4 x + 7 = 16

In this problem, we're asked to solve the exponential equation. And we're given 4 to the power of x+7 is equal to 16. Now looking at these two bases, I have a base of 4 and then I have this 16 on the other side. So I want to think how I can rewrite these to be the same base. Now I know that 16 is actually just 4 squared, so it's actually pretty simple. I can write this as a 4 to the x plus 7 without having to change that base and that is equal to 4 squared. Now you could also both write these as a base of 2, but that would be a little bit more complicated, and they're already the same base now. So since these both have a base of 4, I can go ahead and just set those exponents equal to each other, x+2. So I have x+7 is equal to 2. And now I can just solve this basic linear equation. Now isolating x here, I wanna subtract 7 from both sides, which gives me x is equal to negative 5. So if I plug that negative 5 back into my exponential equation, I know that it would result in a true statement. Thanks for watching, and I'll see you in the next one.

0. Functions

Exponential & Logarithmic Equations

Calculus

0. Functions

Exponential & Logarithmic Equations

Struggling with Calculus?

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Multiple Choice

Solve the exponential equation.
$4^{x+7}=16$

$2$

$5$

$9$

$-5$

Verified step by step guidance

First, identify the structure of the exponential equation. The equation given is $ 4^{x+7} = 16 $.

Recognize that 16 can be expressed as a power of 4. Specifically, $ 16 = 4^2 $. This allows us to rewrite the equation as $ 4^{x+7} = 4^2 $.

Since the bases are the same, you can set the exponents equal to each other. This gives us the equation $ x+7 = 2 $.

Solve for $ x $ by isolating it on one side of the equation. Subtract 7 from both sides to get $ x = 2 - 7 $.

Simplify the expression to find the value of $ x $. This results in $ x = -5 $.

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