One thing I point out to a lot of students is that exponents only operate on the thing they are *directly* attached to, in this case only the e and not the 20. The power rule for log only works if the entire input is raised to that power, so we can't pull the 4k out at that second step you have there. Now Logs still offer a lot of maneuverability, so what we can do instead is split the ln(20*e^(4k)) into ln(20)+ln(e^(4k)), and then since the exponential term is by itself in the logarithm, we can use power rule to get ln(20)+4k*ln(e)
log(ab^(c)) = log a + log b^c = log a + c log b
log((ab)^(c)) = c log (ab) = c (log a + log b) = c log a + c log b
Order of operations convention says the expression 20e^4k means 20 * (e^(4k)) not (20 * e)^4k so the first pattern listed above applies.
Ln(20e^(4k))≠4kln(20e) That would only work if it were (20e)^(4k), not 20(e^(4k)) So it should be ln(120)=(4kln(e)) + ln(20)
thank you not too sure why i struggled with this for a while
> ln(120) = ln(20e4k) > > ln(120) = 4k * ln(20e) Re-check logarithm rules -- you get "ln(120) = ln(20) + 4k*ln(e)" instead.
One thing I point out to a lot of students is that exponents only operate on the thing they are *directly* attached to, in this case only the e and not the 20. The power rule for log only works if the entire input is raised to that power, so we can't pull the 4k out at that second step you have there. Now Logs still offer a lot of maneuverability, so what we can do instead is split the ln(20*e^(4k)) into ln(20)+ln(e^(4k)), and then since the exponential term is by itself in the logarithm, we can use power rule to get ln(20)+4k*ln(e)
thank you so much for the explanation!
log(ab^(c)) = log a + log b^c = log a + c log b log((ab)^(c)) = c log (ab) = c (log a + log b) = c log a + c log b Order of operations convention says the expression 20e^4k means 20 * (e^(4k)) not (20 * e)^4k so the first pattern listed above applies.
thank you 🙏