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

Business Calculus

0. Functions

Exponential & Logarithmic Equations

Struggling with Business Calculus?

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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

Rewrite the equation in terms of base 10. Recall that 100 can be expressed as 10^2. So, rewrite 100^x as (10^2)^x, which simplifies to 10^(2x). The equation becomes 10^(2x) = 10^(x+17).

Since the bases are the same (both are base 10), you can set the exponents equal to each other. This gives the equation 2x = x + 17.

Solve for x by isolating it. Subtract x from both sides of the equation: 2x - x = 17.

Simplify the left-hand side to get x = 17.

Verify the solution by substituting x = 17 back into the original equation to ensure both sides are equal.

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Struggling with Business Calculus?

Solve the exponential equation.100x=10x+17100^{x}=10^{x+17}100x=10x+17

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Exponential & Logarithmic Equations practice set

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