How do you solve #log 5 = (x+1) log 4#?

1 Answer
Dec 31, 2015

This would be same as solving a multi-step linear equation only the coefficients and constants are logarithmic numbers. Step by step working is shown below.

Explanation:

#log(5)=(x+1)log(4)#

Let us distribute #(x+1)log(4)#

#log(5) = xlog(4)+log(4)#

What we see here is not different from any multi-step linear equation .
Say if you had no log and the equation was #5=4x+4# you would have found it easy!

We shall use the same trick to solve it, but with the log present.

#log(5)=xlog(4)+log(4)#
First subtract #log(4)# on both the sides, this is done to isolate the #x#

#log(5)-log(4) = xlog(4)#

#log(5/4)=xlog(4)# Note : #log(P) - log(Q) = log(P/Q)#

Now we divide both sides by #log(4)# to isolate #x#

#log(5/4)/log(4) = x#

The exact solution is #x=log(5/4)/log(4)#

If you want an decimal approximation then you can use a calculator to find it.