Solve the logarithmic equation. log ( x + 2 ) + log 2 = 3 \log\left(x+2\right)+\log2=3 lo g ( x + 2 ) + lo g 2 = 3

In this problem, we're asked to solve the log equation. And the equation we're given is log of x plus 2 plus log of 2 is equal to 3. Now because I have this constant 3 here, I know that I can't just have 2 logs with the same base that I'm going to be able to just set things equal to each other. So I know that I'm going to have to use my steps here. Now, the first thing I want to do is to isolate my log expression. Now, looking at this, since I have these 2 logs here, I want to go ahead and condense this down into one log to make this a little bit more simple. Now these have the same base and they're being added together, so that tells me that I can go ahead and use the product rule here. So using the product rule, this becomes a log multiplying these together, 2 times x plus 2. Now we can actually further simplify this down by distributing this 2 in. So this becomes log of 2x+4. And that's equal to that number that I have on the other side, 3. Now that I've isolated my log expression, I just have one log expression by itself, I can go ahead and move on to step 2 and put this into exponential form. Now, I know that this log is just the common log, which means that it's really a log base 10. So starting with our base here to put this in exponential form, starting with that base, raising it to the power of what's on the other side of the equal sign, so 10 to the power of 3, And then coming back around to that other side of the equal sign, 10 to the power of 3 is equal to 2x+4. Now we're in that exponential form. We've completed step 2, and we can go ahead and just solve for x. Now this 10 cubed, I can go ahead and pull this out to simplify it because 10 cubed is just 10 times 10 times 10, which I know is just a1000. So this is really a1000 is equal to 2x+4. Now to solve for x here, I want to go ahead and subtract 4 from both sides, canceling over here leaving me with 996 is equal to 2 x. Now one final step in isolating x is to divide both sides by 2 canceling it over here and leaving me with my final answer that x is equal to 498. So this is my final answer, but I do want to perform one final check here to make sure that this actually is my solution by plugging it back into the original argument of my log or what I was taking the log of. So I was originally taking the log of 2x+4. So I want to plug that 498 back into it in order to check. So this is 2 times 498+4. And 2 times 498 is that 996 that we originally had. And then a +4 actually just takes us back to 1,000, which is greater than a 0. So we are done, and this is our final answer, x is equal to 498. Let me know if you have any 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 logarithmic equation.
$\log\left(x+2\right)+\log2=3$

$498$

$1998$

$6$

No Solution

Verified step by step guidance

Rewrite the logarithmic equation using the properties of logarithms. Combine the terms on the left-hand side using the product rule: $ \log(a) + \log(b) = \log(a \cdot b) $. This gives $ \log((x+2) \cdot 2) = 3 $.

Simplify the expression inside the logarithm: $ \log(2(x+2)) = 3 $.

Rewrite the equation in its exponential form to eliminate the logarithm. Recall that $ \log_b(a) = c $ implies $ a = b^c $. Here, the base is assumed to be 10, so $ 2(x+2) = 10^3 $.

Simplify the exponential equation: $ 2(x+2) = 1000 $. Divide both sides by 2 to isolate $ x+2 $: $ x+2 = 500 $.

Solve for $ x $ by subtracting 2 from both sides: $ x = 500 - 2 $. Verify the solution by substituting it back into the original equation to ensure it satisfies the logarithmic equation.