Question

How do you solve $ {{10}^{x+4}}=1000 $ ?

Answer 1

Hint: In this question, we need to solve the given equation to find the value of x. For this, we will take logarithm with base 10 on both sides and then apply some logarithm and exponential properties to find a linear equation of x. Solving that equation will give us the required value of x. We will use the following properties,
\[\begin{align}
  & \left( i \right){{\log }_{b}}{{a}^{m}}=m{{\log }_{b}}a \\
 & \left( ii \right){{\log }_{a}}a=1 \\
 & \left( iii \right){{a}^{n}}=\underset{n\text{ times}}{\mathop{\underline{a\times a\times a\times a\times \cdots \cdots \times a}}}\, \\
\end{align}\]

Complete step by step answer:
Here we are given the equation as $ {{10}^{x+4}}=1000 $ . We need to find the value of x which satisfies this equation. For this, let us take the logarithm with base 10 on both sides we get, $ {{\log }_{10}}{{10}^{x+4}}={{\log }_{10}}1000 $ .
We know that, $ {{\log }_{b}}{{a}^{m}} $ can be written as $ m{{\log }_{b}}a $ . So, left side of the equation can be written in this way we get $ \left( x+4 \right){{\log }_{10}}10={{\log }_{10}}1000 $ .
As we know that, $ {{\log }_{b}}b=1 $ so in the left side of the equation we have b = 10 we get \[\left( x+4 \right)1={{\log }_{10}}1000\Rightarrow x+4={{\log }_{10}}1000\].
Now we need to solve the right side of the equation. We know, 1000 can be written as $ 10\times 10\times 10 $ . And also, we know \[{{a}^{n}}=\underset{n\text{ times}}{\mathop{\underline{a\times a\times a\times a\times \cdots \cdots \times a}}}\,\] so we have $ 1000=10\times 10\times 10={{10}^{3}} $ . So our equation becomes \[x+4={{\log }_{10}}{{10}^{3}}\].
Using the logarithmic property \[{{\log }_{b}}{{a}^{m}}=m{{\log }_{b}}a\] we get \[x+4=3{{\log }_{10}}10\].
Using the property $ {{\log }_{b}}b=1 $ we get $ x+4=3\cdot 1\Rightarrow x+4=3 $ .
Taking 4 to the other side we get $ x=3-4\Rightarrow x=-1 $ .
Therefore the required value of x is -1.

Note:
 Students should keep in mind all the logarithmic properties. Students can solve this sum using the following method also, we have $ {{10}^{x+4}}=1000 $ .
We know $ 1000={{10}^{3}} $ so we get $ {{10}^{x+4}}={{10}^{3}} $ .
By the law of indices, we know that, if the base is the same on both sides then the exponent is also the same. Therefore, x+4 = 3, solving it we get x = -1.
Take care of the signs while solving the equation.