I see all the time kids asking about the same questions so I will create this as to reference when basic methodology is needed to be taught

To start off basic exponent rules:


a^{x}\cdot{a^{y}}=a^{x+y}...NOTE a^{x}\cdot{b^{y}}\ne{(ab)^{x+y}}


Using this in conjunction with the second rule we can see that



Now I will move onto logarithmic properties

NOTE: log_e(x)=\ln(x) is used in lieu of other logarithims for its shortness....these rules apply to all logarithims...and for knowledge

lg(x)=log_2(x) and log(x)=log_{10}(x)





A very important one for solving equations as well as using a calculator is the change of base theorem which states


Most useful example is log_a(b)=\frac{\ln(b)}{\ln(a)}

another thing you should know is




Now I will go over the way to solve two basic logarithmic equations

Case 1...this is one with a pre-existing log


first combien the logs to get


we want to isolate the x so we introduce the exponential function( e^{x}) to eliminate the log

so we have e^{\ln(a^2)}\Rightarrow{a^2=e^8}

taking the squareroot of both sides and using rules 5 and 6 in the exponent section we see the answer is a=e^{4}

Case 2....introducing logs to eliminate exponents...


First thing we want to do is introduce a log to elminate the exponent...the log we introduce is always of same base as the exponent in this case 8..so we have


at this point I think it is prudent to discuss extraneous solutions

\log_a(x) is defined only for positive values for x...so you might get an algaebraic solution that works but it makes a log undefined..these solutiong are to be discarded

any questions just ask