If \( \ln (x-8)-\ln (6)=\ln (7) \) then \( x=\square \)

To solve the equation \( \ln (x-8)-\ln (6)=\ln (7) \), we can use properties of logarithms. First, use the quotient property of logarithms: \[ \ln\left(\frac{x-8}{6}\right) = \ln(7) \] By exponentiating both sides to eliminate the logarithm, we get: \[ \frac{x-8}{6} = 7 \] Now, multiply both sides by 6: \[ x-8 = 42 \] Finally, adding 8 to both sides, we find: \[ x = 50 \] So, \( x = 50 \). --- Did you know that logarithms have been around since the early 17th century? John Napier, a Scottish mathematician, introduced logarithms as a way to simplify complex calculations, such as multiplication and division, by transforming them into addition and subtraction! Now, we can calculate everything quickly with our calculators, thanks to the magic of logarithms! In real-world scenarios, logarithms aren't just useful in math class—they're essential in fields like finance and biology! For example, in finance, logarithmic scales can help model exponential growth of investments, and in biology, they can describe population growth or the decay of substances. Who knew math could take us so far!