Question

Find the value of $\log 4.5$.

Answer 1

Hint: Logarithmic function is inverse function of exponential function. Base of $\log $ is $10$. For this question, try to express the given term in the form of $\log 2,log3$ as the standard values of $\log 2$ and $log3$ are known up to three places of decimals. This can be done by first writing 4.5 as a fraction and then using the logarithmic properties $\log \left( \dfrac{m}{n} \right)=\log m-\log n$ and $\log {{m}^{a}}=a\log m$.

Complete step-by-step answer:
We have to find the value of $\log 4.5$.
The base of $\log $ is $10$.
We know the standard value of $\log 2$ and $\log 3$ .
So we will try to express the term $\log 4.5$ in the form of $\log 2$ and $\log 3$.
Now we have to express $4.5$ as a product, quotient and/or power of $2$’s $3$’s and/or power of $10$.
In the decimal number $4.5$ it has one place of decimal. So to express $4.5$ in fraction form, the numerator will be $45$ and denominator will be $10$.
Therefore we will get it as $4.5=\dfrac{45}{10}$.
Dividing both numerator and denominator by $5$ we get $4.5=\dfrac{9}{2}$.
Therefore we have $\log 4.5=\log \dfrac{9}{2}$.
We know the quotient logarithmic property that $\log \left( \dfrac{m}{n} \right)=\log m-\log n$.
Using the above formula we get $\log 4.5=\log 9-\log 2$.
Now as $9={{3}^{2}}$ so $\log 9=\log {{3}^{2}}$.
Therefore we get $\log 4.5=\log {{3}^{2}}-\log 2$.
We know the product logarithmic property that $\log {{m}^{n}}=n\log m$.
Using this formula of logarithm we have $\log 4.5=2\log 3-\log 2$.
The standard value of $\log 3$ is $0.477$ and standard value of $\log 2$ is $0.301$ up to three places of decimals.
Putting these values we get $\log 4.5=2\times 0.477-0.301$.
Further simplifying we get $\log 4.5=0.954-0.301$.
Solving the rest subtraction we get $\log 4.5=0.653$.
Hence the value of $\log 4.5=0.653$ up to three places of decimals.
This is the required solution.

Note: In this problem the main key is to express the term in the logarithm of $2$’s $3$’s and/or power of $10$. For that students have to use logarithmic properties, so students must be aware of those. The probable mistake that students can make is using the wrong logarithmic property and then complicating the solution. Always write a step by step solution to avoid the calculation error. Students must learn the values of log 2, log 3 and log 5 as they are very helpful in solving such questions.