Question

How do you solve \[\log x+\log 5=2\]?

Answer 1

Hint: We can see that the question involves the logarithm function, so we will be using the properties of logarithms to solve for \[x\]. We will first convert the 2 present in RHS in terms of logarithm function, that is, \[2={{\log }_{10}}{{10}^{2}}={{\log }_{10}}100\]. We will next use the logarithm property which is \[\log a+\log b=\log ab\] and we will get the expression as \[\log 5x=\log 100\]. Then, on comparing the expression on either side of the equality, we will get the value of \[x\].

Complete step by step solution:
According to the given question, we have a logarithm based question, which we have to solve for \[x\]. We will be using the properties of the logarithm function to solve for the same.
We first need to understand that logarithm function and numbers cannot be computed together. We will have to convert the entire expression in terms of logarithm or in terms of numbers to start solving for \[x\]. In this given question, we can see it will be better if we convert all the terms in the expression in terms of logarithm function, as only one term on the RHS is a number.
So, we will write 2 in the RHS in terms of logarithm function,
We know that,
\[2={{\log }_{10}}{{10}^{2}}\]
As \[{{\log }_{10}}10=1\]
So, we have,
\[2={{\log }_{10}}{{10}^{2}}={{\log }_{10}}100\]
The new expression we have is,
\[\log x+\log 5=\log 100\]----(1)
Now, we will be using the property of the logarithm function which is,
\[\log a+\log b=\log ab\]---(2)
Applying the equation (2) in equation (1), we get,
\[\Rightarrow \log 5x=\log 100\]
We can see that the above expression has log function on both sides of the equality, so we cancel it out, and we have,
\[\Rightarrow 5x=100\]
We have the value of \[x\] as,
\[\Rightarrow x=20\]
Therefore, the value of \[x\] is \[20\].

Note: The logarithm properties we have:
1) \[\log a+\log b=\log ab\]
2) \[\log a-\log b=\log \dfrac{a}{b}\]
While using the properties, the values should be substituted correctly. Also, be careful of the base of the logarithm function and check whether it is 10 or exponential and accordingly solve the expression.