Solving Longarithmic Equations

log₇ 4 - log₇ (x-4) = log₇ 41

We can write the left side as

log₇ [ 4 / (x - 4)  ]   = log₇ 41

Since the logs are the same on each side, we can ignore these and solve the remaining equation

4 /(x - 4)  = 41      multiply both sides by (x - 4)

4 = 41 (x - 4)    simplify

4 = 41x - 164    add 164 to both sides

4 + 164   = 41x

168 = 41x     divide both sides by 41

168 / 41  = x

Solving Longarithmic Equations

log₇ 4 - log₇ (x-4) = log₇ 41

\(\begin{array}{|rcll|} \hline \log_7{(4)} - log_7{(x-4)} &=& log_7{(41)} \qquad | \qquad + log_7{(x-4)} \\ \log_7{(4)} &=& log_7{(41)} + log_7{(x-4)} \\ && \begin{array}{|rcll|} \hline \qquad \log(a) + \log(b) &=& \log(a\cdot b) \\ \hline \end{array} \\ \log_7{(4)} &=& log_7{(~41\cdot (x-4)~)} \\ 4 &=& 41\cdot (x-4) \\ 4 &=& 41x-4\cdot 41 \\ 41x &=& 4 + 4\cdot 41 \\ 41x &=& 4\cdot (1 + 41) \\ 41x &=& 4\cdot 42 \qquad | \qquad :41 \\ x &=& 4\cdot \frac{42}{41} \\ x &=& 4\cdot 1.02439024390 \\ \mathbf{x} &\mathbf{=} & \mathbf{4.09756097561} \\ \hline \end{array}\)