Question

Consider the equation log (3x-1)=log base 2 of 8. Describe the steps you would take to solve the equation, and state what 3x-1 is equal to.

Answer 1

The bases are not the same, so you cannot set 3x - 1 equal to 8.

You can evaluate the logarithm on the right side of the equation to get 3.

You can use the definition of a logarithm to write 3x - 1 = 1000.

Answer 2

Answer:  The solution of the given equation is $x=333\dfrac{2}{3}$ and the value of (3x - 1) is 1000.

Step-by-step explanation:  We are given to describe the steps in solving the following logarithmic equation:

$\log(3x-1)=\log_28.$

Also, we are to find the value of $(3x-1).$

We will be using the following logarithmic properties:

$(i)~\log_ba=x~~~\Rightarrow a=b^x,\\\\(ii)~\log_ba=\dfrac{\log a}{\log b},\\\\(iii)~\log a^b=b\log a.$

We note here that if the base of logarithm is not mentioned, then we assume it to be 10.

The solution is as follows:

$\log(3x-1)=\log_28\\\\\Rightarrow \log(3x-1)=\dfrac{\log8}{\log2}\\\\\Rightarrow log(3x-1)=\dfrac{\log2^3}{\log2}\\\\\Rightarrow \log(3x-1)=\dfrac{3\log2}{\log2}\\\\\Rightarrow \log(3x-1)=3\\\\\Rightarrow 3x-1=10^3\\\\\Rightarrow 3x-1=1000\\\\\Rightarrow 3x=1001\\\\\Rightarrow x=\dfrac{1001}{3}\\\\\Rightarrow x=333\dfrac{2}{3}.$

Thus, the solution of the given equation is $x=333\dfrac{2}{3}$ and the value of (3x - 1) is 1000.