My book tells me to solve each exponential equation. I am given a lot of problems such as
6^x+1 = 4^2x-1
where I use In in the problem to help get my answer. However, in the same section, the book starts giving me problems such as
3(2)^x-2 +1=100
and .05(1.15)^x =5 etc.
Where I apparently have to use log in the equations instead of using In like I did in all the other problems. Why is this? Like, how will I know weather to use In or log to find my answer? Is there a reason why you use log instead?
It makes no difference if you use ln or log unless the exercise tells you to. If you are allowed a calculator then it makes no difference.
In the example above:
3*2^(x-2)+1=100 3*2^(x-2)=99 2^(x-2)=33
If we use ln we get: ln2^(x-2)=ln33
(x-2)ln2=ln33 x= 2+ln33/ln2=7.044
If we used log with base 2 we would get: x-2=log(base2)33 so x= 2+log(base2)33 =7.044
No difference!
When dealing with exponential equations, the choice between using natural logarithm (ln) or logarithm (log) depends on the base of the exponential expression in the equation.
If the base of the exponential expression is e (approximately equal to 2.71828), then you use ln to solve the equation. This is because ln is the inverse function of the natural exponential function, e^x, and they "cancel" each other out.
On the other hand, if the base of the exponential expression is something other than e, such as 2 or 10, then you use log to solve the equation. Logarithms are the inverse functions of exponential functions with bases other than e.
In the first problem you mentioned, 6^(x+1) = 4^(2x-1), the bases of both exponential expressions are not e. Therefore, you should use log to solve it.
In the second problem, 3(2)^(x-2) + 1 = 100, the base of the exponential expression is 2, which is not e. Therefore, you should also use log to solve it.
Similarly, in the third problem, 0.05(1.15)^x = 5, the base of the exponential expression is 1.15, which is not e. Hence, you should use log to solve it as well.
To summarize, use ln when the base is e and use log when the base is other than e. The choice of ln or log is based on the mathematical properties of logarithmic and exponential functions, and it ensures that you obtain accurate solutions to the equations.