Solve the exponential equation. 10 0 x = 1 0 x + 17 100^{x}=10^{x+17} 10 0 x = 1 0 x + 17

In this problem, we're asked to solve the exponential equation. And here, we're given the equation 100 to the power of x is equal to 10 to the power of x plus 17. Now, I want to rewrite these with like bases. And the very first thing I notice is that 100 is just 10 squared and my base on the other side is already 10. So this would be a great change to make in order to get my like bases. So this is 10 squared to the power of x, and that's equal to 10 to the power of x plus 17. Now using my exponent rules, I wanna go ahead and rewrite this with these exponents multiplied. So this is really just 10 to the power of 2x, and that's equal to 10 to the power of x plus 17. Now I have like bases on either side. They're both 10, so I can go ahead and take my exponents, 2x and x plus 17, and set them equal to each other. So I get 2x is equal to x plus 17, and we're left to solve this linear equation. Now I want to go ahead and move my x's all to the same side since I have a variable on both sides right now. So subtracting this x from either side, it will cancel on this side, leaving me with just that 17. And then 2x minus x is just 1x. So I'm actually completely done here. And my answer is x is equal to 17. Now if I were to plug this 17 back into my original equation, I'm going to end up with a true statement. Hopefully, that makes sense. Let me know if you have questions.

0. Functions

Exponential & Logarithmic Equations

Calculus

0. Functions

Exponential & Logarithmic Equations

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Multiple Choice

Solve the exponential equation.
$100^{x}=10^{x+17}$

$17$

$34$

$8.5$

$0$

Verified step by step guidance

Start by recognizing that the equation is in the form $100^x = 10^{x+17}$. This suggests that we can use properties of exponents to simplify the equation.

Express $100$ as $10^2$ so that the equation becomes $(10^2)^x = 10^{x+17}$.

Apply the power of a power property $(a^m)^n = a^{m \cdot n}$ to rewrite the left side as $10^{2x}$. The equation now is $10^{2x} = 10^{x+17}$.

Since the bases are the same, set the exponents equal to each other: $2x = x + 17$.

Solve the linear equation $2x = x + 17$ by subtracting $x$ from both sides to get $x = 17$.

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