Question

Solve. Show your work.log3 (2x + 1) = log3 (6x – 23)

Answer 1

Answer:

$x=6$

Explanation:

To solve this, we need to use two properties of logarithms.

The first one is the difference of logarithms with same base:

$\log_a(x)-\log_a(y)=\log_a(\frac{x}{y})$

The second one is the definition of logarithm:

$\log_aN=x\Leftrightarrow N=a^x$

Then, we are given the equation:

$\log_3(2x+1)=\log_3(6x-23)$

Then:

$\operatorname{\log}_3(2x+1)-\operatorname{\log}_3(6x-23)=0$

Applying the first property, for difference of logs:

$\operatorname{\log}_3(\frac{2x+1}{6x-23})=0$

Now we apply the definition of log:

$\log_3(\frac{2x+1}{6x-23})=0\Leftrightarrow\frac{2x+1}{6x-23}=3^0$

And now we can solve as any rational equation:

$\begin{gathered} \frac{2x+1}{6x-23}=3^0 \\ . \\ \frac{2x+1}{6x-23}=1 \\ . \\ 2x+1=1(6x-23) \\ 2x+1=6x-23 \\ 1+23=6x-2x \\ 24=4x \\ . \\ x=\frac{24}{4}=6 \end{gathered}$

Thus, the answer is x = 6