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Question: How do you solve \( {2^x} = 15 \) ?...

How do you solve 2x=15{2^x} = 15 ?

Explanation

Solution

Hint : To find the answer to the above question, our first approach should be to separate the unknown variable on one side of the equation. To do so we will take logs on both sides of the equation. This step will bring us a point where the unknown variable will be equal to a value of log⁡xlog⁡y\dfrac{{\log x}}{{\log y}} . Here, we will use the logarithm property that log⁡xlog⁡y=log⁡yx\dfrac{{\log x}}{{\log y}} = {\log _y}x to simplify the answer to the best possible version. Also, calculators can be used in the final step to get the values of log in order to get the most precise answer.

Complete step-by-step answer :
The given equation in the question is
2x=15{2^x} = 15
Now, in order to simplify this equation more to find the answer we have to separate the x and keep it on one side of the equation and the rest on the other
So, to do that we will take log on both the sides
Therefore,
log⁡2x=log⁡15   \log {2^x} = \log 15 \;
Now using the property of log which says log⁡xy=ylog⁡x{\log ^{{x^y}}} = y\log x
log⁡2x\log {2^x} becomes, xlog⁡2x\log 2
Hence,
xlog⁡2=log⁡15 ⇒x=log⁡15log⁡2   x\log 2 = \log 15 \\\ \Rightarrow x = \dfrac{{\log 15}}{{\log 2}} \;
Now, another property of logarithm states that log⁡xlog⁡y=log⁡yx\dfrac{{\log x}}{{\log y}} = {\log _y}x
Hence,
x=log⁡15log⁡2 =log⁡215   x = \dfrac{{\log 15}}{{\log 2}} \\\ = {\log _2}15 \;
Therefore the solution of 2x=15{2^x} = 15 is x=log⁡215x = {\log _2}15 .
So, the correct answer is “ x=log⁡215x = {\log _2}15 ”.

Note : One can directly put values of log 15 and log 2 to calculate the value of x in the steps above. But in cases when you don’t know the value of the logarithmic numbers then just try to simplify the numbers as much as possible. The most simplified version of the numbers would suffice for the final answer of the question