I am trying to figure out how to solve a math problem and am awful at math so go easy on me. We are working on logarithms. I am given the answer though I'd like to know how to solve it on my calculator.
$\log 1.8 = \log 1.0205{^2t}$
$\log 1.8 = 2t * \log 1.0205$
$t = \frac{\log 1.8}{2 \log 1.0205}$
$t = 14.4827$
somehow they figured this out on the calculator and I can't figure out how.
Every time I type in $\log 1.8$ on the calculator I get $.2555$.
If anyone can help a total math dummy figure out how to use a TI-83 plus to figure this out I'd be very grateful. Thank you!
There are two types of logarithms on a calculator - the common logarithm in base $10$, denoted by $\log x$, and the natural logarithm in base $e = 2.7182818...$, denoted by $\ln x.$ Sometimes, $\log x$ will often represent $\ln x$ while the base ten logarithm will be noted as $\log_{10} x$.
When you hit the $\log$ button on your TI-83, the calculator computed your equation using the common logarithm. That is, $$t = \dfrac {\log 1.8}{2 \log 1.0205} = \dfrac {0.2553}{0.0176} = 14.4827.$$
When you hit the $\ln$ key, you're computing using natural logarithms, so you get $$t = \dfrac {\ln 1.8}{2 \ln 1.0205} = \dfrac {0.5878}{0.0406} = 14.4827.$$
Note that the answers for each are equal.
CAVEAT: In order to get similar results, you must use the same base. Using two different bases will produce different results, that is $$t = \dfrac {\ln 1.8}{2 \log_{10} 1.0205} = \dfrac {0.5878}{0.0176} = 33.3477.$$ while $$t = \dfrac {\log_{10} 1.8}{2 \ln 1.0205} = \dfrac {0.2553}{0.0406} = 6.2898.$$