So as we saw in the introduction into logarithms, logarithms are made up of exponential equations and are used as a means to solve them.
For example, if we see a question like 2^x=8, we turn that into logarithmic form as Log2(8)=x. Next, we use the change of base formula and we get Log(8)/Log(2)=x which when plugged into our calculators, we get 3. Therefore, x=3 which as we know, is true that 2^3 is 8.
Another Example would be if we have 4^x=1024. Once again we convert it into logarithmic form, Log4(1024)=X and solve by using the change of base formula to give us Log(1024)/Log(4)=X which will give us 5 as our answer. We can verify once again and see that 4^5 indeed does give us 1024. One small thing to note though, we will not always be faced with exact values. For instance, if the 1024 was instead something like 2000, then results would be different and give us 5.482892142. we round off to 2 decimal places which will give us approximately 5.48. While not an exact value, we should always round of to 2 decimal places or unless asked for otherwise.
However, exponential equations may not always be as simple as these ones. For instance, we may have questions like 2^x+1=10^2.
To handle these equations, we must always turn them into logarithmic forms like Log(2^x+1)=Log(10)^2.
next, we must use the power rule to simplify these terms which gives us the following. (x+1)Log(2)=2Log(10)
continuing to simplify, we must use the distributive method to separate the 1 and the x like this: x Log(2)+Log(2)=2Log(10).
we then isolate the x by subtracting the log(2) to take it to the other side. x Log(2)+Log(2)-Log(2)=2Log(10)-Log(2).
Following our procedures to isolating the x, we divide by log(2) on both sides which will isolate the x and give us:
x=2Log(10)-Log(2) all over Log(2).
Our final answer after plugging it into the calculator will be approximately, x=5.64(2d.p.)
We may get variables on both sides but we follow the same procedures as seen here.
Exponential equations require the use of most logarithm rule which can make the experience either a pleasant one or a stressful situation. These are the best way to get to understand and memorize the different logarithm rules.