Example Solve out each of the following equations.
7^{x} = 9
Solution
Okay, although we say above that if we contained a logarithm in front the left side we could obtain the x out of the exponent. That's simple enough to do. We'll only put a logarithm in front of the left side. But, if we put a logarithm there we also have to put a logarithm in front of the right side. Commonly this is referred to as taking the logarithm of both sides.
We can employ any logarithm that we'd like to thus let's try the natural logarithm.
ln 7^{x} = ln 9
x ln 7 = ln 9
Now, we have to solve for x. it is easier than it looks. If we had 7 x = 9 then we could all solve out for x simply by dividing both of the sides by 7. It works in accurately the same manner here. Both ln7 & ln9 are only numbers. Admittedly, it would take a calculator to find out just what those numbers are, however they are numbers & so we can do the similar thing here.
x ln 7/ ln 7 = ln 9/ ln 7
x = ln 9/ ln 7
Now, technically i.e. the exact answer. Though, in this case usually it's best to get a decimal answer so let's go one step further.
x = ln 9 / ln 7= 2.19722458 /1.94591015= 1.12915007
Note down that the answers to these are decimal answers more frequently than not.
Also, be careful to not make the following mistake here.
1.12915007 = ln 9/ ln 7 ≠ ln ( 9 /7)=0.2513144283
The two are apparently different numbers.
At last, let's also utilizes the common logarithm to ensure that we get the same answer.
log 7^{x} = log 9
x log 7 =log 9
x =log 9/ log 7 = 0.954242509 /0.845098040= 1.12915007
Thus, sure enough the similar answer. We can use either logarithm, even though there are times while it is more convenient to employ one over the other.