Solve the exponential equation. 900 = 1 0 x + 17 900=10^{x+17} 900 = 1 0 x + 17

In this problem, we're asked to solve the exponential equation. And here we're given the equation 900 is equal to 10 to the power of x plus 17. Now let's go ahead and get started with step 1 and isolate our exponential expression. But our exponential expression is actually already isolated. It's this 10 to the power of x plus 17, so we're good on step 1. We can move on to step 2. Now in step 2, we wanna check if there is a base 10 or not a base 10, and it looks like, yes, we do have a base 10, so we're going to just use log, the common log. So taking the log of both sides, I'm going to get log of 900 is equal to log of 10 to the power of x plus 17. Now that step 2 is done, we can move on to step 3 and use our log rules to get x out of the exponent. So in order to do that, I'm going to use the power rule and pull this x plus 17 to the front of my log. So this is log of 900 is equal to x plus 17 times log of 10. Now we can go ahead and simplify this further using log rules because we know that log of 10 is really just 1. So this goes away. It's just 1, and this is just log of 900 is equal to x plus 17. Now that we've completed step 3, we can go ahead and just solve for x. So solving for x here, I only have one thing to do and that's subtract this 17 from both sides. Now whenever you rewrite this, you need to be careful and use parentheses where needed or you can just put the 17 in at the front. It is up to you, but we just want to make sure that we are not making it seem like that 17 is being subtracted from the 900 because log 900 is something by itself and we're subtracting 17 from that. So this is a log of a 900. I'm gonna go ahead and use parentheses and then minus 17, and that's equal to x. Now if you're just using this in the log form, you can go ahead and stop here. But since we were done with step 4, we can go ahead and approximate this using a calculator. Now, plugging this into your calculator, make sure to use parentheses where appropriate because we're just taking the log of 900 and then subtracting 17 from that. Now, when you do that, you'll end up getting negative 14.05, rounding to that nearest hundredth place, and that is equal to x. And here is our final answer. Thanks for watching, and I'll see you in the next one.

0. Functions

Exponential & Logarithmic Equations

Calculus

0. Functions

Exponential & Logarithmic Equations

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Multiple Choice

Solve the exponential equation.
$900=10^{x+17}$

$x = - 14.05$

$x equals 2.95$

$x equals 0.17$

$x equals 1.72$

Verified step by step guidance

Start by isolating the exponential term. The equation is 900 = 10^{x+17}. To isolate 10^{x+17}, subtract 17 from both sides to get 900 - 17 = 10^{x}.

Simplify the left side of the equation: 900 - 17 = 883. So the equation becomes 883 = 10^{x}.

To solve for x, take the logarithm of both sides. Use the base 10 logarithm because the base of the exponential is 10. This gives you log_{10}(883) = x.

Calculate log_{10}(883) using a calculator or logarithm table to find the value of x.

Verify your solution by substituting the value of x back into the original equation to ensure both sides are equal.