Solve the logarithmic equation algebraically.Approximate the result to three decimal places.

Question

Solve the logarithmic equation algebraically.Approximate the result to three decimal places. ​  ​

Quizplus · Accepted Answer

We have:

$log_2(3x + 5)-log_2(x-1) = 3$

Using the logarithmic identity:

$log_b\frac{a}{c} = log_ba - log_bc$

we can simplify the left-hand side as follows:

$log_2\frac{3x+5}{x-1} = 3$

Using another logarithmic identity:

$b^{log_bc} = c$

we can rewrite the equation as:

$\frac{3x+5}{x-1}=2^3=8$

Next, we can solve for x by multiplying both sides by x-1 and simplifying:

$3x+5=8(x-1)$

$3x+5=8x-8$

$5x=13$

$x=\frac{13}{5}$

Finally, we can verify that this solution works by substituting it back into the original equation and checking that both sides are equal:

$log_2(3\cdot \frac{13}{5} + 5)-log_2(\frac{13}{5}-1) = log_2\frac{44}{5} - log_2\frac{8}{5}$

$= log_2\frac{44}{8} = log_2\frac{11}{2} = 3$

Therefore, the answer is E, 11eacce6_1fe8_6669_9f49_d3cea01320fd_TB7896_11.

Solve the Logarithmic Equation Algebraically