Solve the logarithmic equation. log 3 ( 3 x + 9 ) = log 3 5 + log 3 12 \log_3\left(3x+9\right)=\log_35+\log_312 lo g 3 ( 3 x + 9 ) = lo g 3 5 + lo g 3 12

In this problem, we're asked to solve the log equation. And here, the equation we're given is a log base 3 of 3x plus 9 is equal to log base 3 of 5 plus log base 3 of 12. Now the very first thing I notice about all of these logs is that they all have this same base of 3, which tells me that I can likely simplify this down into a log base 3 on one side, log base 3 on the other, and just set what I'm taking the log of equal to each other. So let's go ahead and try to simplify this down into a single log of the same base on both sides. Now I don't have any work to do on this side. This is just log base 3 of 3x+9. And then on the other side, I see that I have this addition happening, which tells me that I can go ahead and use the product rule and turn this addition into the multiplication inside of a single log. So this is all equal to log base 3 of 5 times 12, which is just 60. So this becomes a log base 3 of 3x+9 is equal to the log base 3 of 60. Now since these logs have the same base, I can just take what I'm taking the log of and set them equal to each other. So let's go ahead and do that over here. We have 3x+9 is equal to 60. And now we just have a basic linear equation to solve. So I'm gonna go ahead and subtract 9 from both sides. That will cancel over here, leaving me with 3x is equal to 51. 60 minus 9 gives me that 51. And we have one final step here, which is to isolate x completely and divide both sides by 3. Now that will cancel out over here, leaving me with x is equal to 51 divided by 3 which is just 17. So my final answer is that x is equal to 17. Thanks for watching, and I'll see you in the next one.

6. Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

Precalculus

6. Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

Struggling with Precalculus?

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Multiple Choice

Solve the logarithmic equation.
$\log_3\left(3x+9\right)=\log_35+\log_312$

No Solution

Verified step by step guidance

Start by using the property of logarithms that allows you to combine the right side of the equation: $ \log_3 5 + \log_3 12 = \log_3 (5 \times 12) $. This simplifies the equation to $ \log_3 (3x + 9) = \log_3 60 $.

Since the logarithms on both sides of the equation have the same base, you can set the arguments equal to each other: $ 3x + 9 = 60 $.

Subtract 9 from both sides to isolate the term with $ x $: $ 3x = 60 - 9 $.

Simplify the right side: $ 3x = 51 $.

Divide both sides by 3 to solve for $ x $: $ x = \frac{51}{3} $.