Solve the logarithmic equation. log 7 ( 6 x + 13 ) = 2 \log_7\left(6x+13\right)=2 lo g 7 ( 6 x + 13 ) = 2

Here we're asked to solve the log equation. And the equation we're given is log base 7 of 6x plus 13 is equal to 2. Now since I just have a single log equal to something, I know that I need to go ahead and use my steps. So let's go ahead and start with step 1 and isolate our log expression. Now our log expression is actually already by itself, so we're good here. And we're completely done with step 1. And we can move on to step 2 and go ahead and put this in exponential form. Now remember, we're always going to start from our base. So starting from that base, circling to the other side of my equal sign, I have 7 squared is equal to circling back over here, 6 x plus 13. So here I have 7 squared is equal to 6 x plus 13. It's in that exponential form, and we can go ahead and move on to step 3 and simply solve for x. Now I want to go ahead and take this 7 squared and multiply that out. 7 squared is just 7 times 7. I know that that's 49. So I have 49 is equal to 6x plus 13. Now solving for x here, the first thing I want to do is subtract 13 from both sides. So doing that cancels over here. 49 minus 13 will give me 36, and that is equal to 6 x. Now one final step here in isolating x, I just need to divide both sides by 6, leaving me with 36 divided by 6. So my final answer is that x is equal to 6. Now we've solved for x but we want to perform that one final check by plugging this into our original equation. So we wanna take what we were taking the log of, that 6x plus 13, and make sure that it is positive with our answer. So plugging this back in, I get 6 times 6 plus 13, which is really just 36 +13, which gets us back to that 49. Now 49 is greater than 0. It is a positive number. So we are completely done, and this does satisfy everything that we need it to. This is our final answer, x is equal to 6. Thanks for watching, and I'll see you in the next one.

0. Functions

Exponential & Logarithmic Equations

Business Calculus

0. Functions

Exponential & Logarithmic Equations

Struggling with Business Calculus?

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Multiple Choice

Solve the logarithmic equation.
$\log_7\left(6x+13\right)=2$

$3$

$19.17$

$6$

No Solution

Verified step by step guidance

Rewrite the given logarithmic equation log_7(6x+13) = 2 in its exponential form. Recall that log_b(a) = c is equivalent to b^c = a. Here, 7^2 = 6x + 13.

Simplify the exponential equation 7^2 = 6x + 13. Calculate 7^2 to get 49, so the equation becomes 49 = 6x + 13.

Isolate the variable x by subtracting 13 from both sides of the equation. This gives 49 - 13 = 6x.

Simplify the left-hand side to get 36 = 6x. Then, divide both sides of the equation by 6 to solve for x, resulting in x = 36/6.

Verify the solution by substituting x back into the original logarithmic equation. If the left-hand side equals the right-hand side, the solution is valid. If not, there is no solution.