Question

How do you solve \[{5^x} = 28\].

Answer 1

Hint: In this question, we have an exponential function. To solve this function, we log both sides (which is called a logarithm function) then we use the formula. The formula is given as below. If the function is \[{\log _b}\left( {{m^n}} \right)\] then it is written as below.
\[{\log _b}\left( {{m^n}} \right) = n.{\log _b}m\]
Here, \[b\] is the base.

Complete step by step solution:
In this question, we used the word logarithm function. First, we know about the logarithm function. The logarithm function is defined as the inverse function to exponentiation. That means that in the logarithm function the number \[x\] is given as the exponent of another fixed number and the base must be raised to produce that number \[x\].
Let’s, the exponential function is denoted as below.
\[ \Rightarrow y = {e^x}\]
It is also expressed as below type.
\[ \Rightarrow y = \exp \left( x \right)\]
The above function is also called a natural exponential function.
We know that \[y = {e^x}\] is a one to one function and we also know that its inverse will also be a function. Then we find the inverse function of \[y = {e^x}\] function. The inverse function of \[y = {e^x}\] is given as below.
\[ \Rightarrow y = {\log _e}x\]
It is also written as the below type.
\[y = \ln x\]
These functions \[y = \log x\] or \[y = \ln x\] are called the logarithm function.
Now come to the question, in the question the function is given as below.
\[ \Rightarrow {5^x} = 28\]
Now we take the log on both sides.
Then,
\[ \Rightarrow \log \left( {{5^x}} \right) = \log 28\]
We know that, if one function is \[{\log _b}m\] and another function is\[{\log _b}n\], which is added together then it is written as below.
\[{\log _b}m + {\log _b}n = {\log _b}\left( {m.n} \right)\]
Here, \[b\] is the base.
In the question, the function is \[\log \left( {{m^n}} \right)\] then is written as the below type.
Then,
\[\log \left( {{m^n}} \right) = n.\log m\]
 We apply this formula for above function.
Then,
\[ \Rightarrow x\log \left( 5 \right) = \log \left( {28} \right)\]
Then, after calculating
\[ \Rightarrow x = \dfrac{{\log \left( {28} \right)}}{{\log \left( 5 \right)}}\]
We know that, \[\log \left( {28} \right) = 1.4471\] and \[\log \left( 5 \right) = 0.6989\]. Put these values in the above.
Then,
\[ \Rightarrow x = \dfrac{{1.4471}}{{0.6989}}\]
\[\therefore x \approx 2.07\]

Therefore, the value of \[x\] is \[2.07\].

Note:
If you have a function \[{a^x} = b\] then first take the log on both sides to find the value of \[x\]. If there is a function \[{\log _b}\left( {{m^n}} \right)\] and you want to find the value of that logarithm function. Then there is a formula for this logarithm function.