Question

show that the value of
$log_{10}3$lies between
$\frac{1}{2} and \frac{2}{5}$

​

Answer 1

Given: $\log_{10}3$

Required answer: $\dfrac{2}{5} <\log_{10}3<\dfrac{1}{2}$

Concepts

• Exponential Inequality

→ It is a kind of inequality involving exponents. Usually, when the bases are the same the comparison of exponents is used to find the solution.

Solution

Let $x=\log_{10}3$. Then, $10^x=3$ by the property of the logarithm.

Lower Bound

Here we are looking for the lower bound. We should keep the base equal to 10.

$\implies 10^{x}=3$

$\implies (10^{x})^{5}=3^{5}$

Now we bring a power of 10 into inequality.

$\implies 10^{5x}=243>100$

$\implies 10^{5x}>10^2$

The bases are the same. Now comparing the exponents,

$\implies 5x>2$

$\implies \boxed{x>\dfrac{2}{5} }$

Upper Bound

Here we are looking for the upper bound. Again, the base should be 10.

$\implies 10^{x}=3$

$\implies (10^{x})^4=3^{4}$

Now we bring a power of 10 into inequality.

$\implies 10^{4x}=81<100$

$\implies 10^{4x}<10^2$

The bases are the same. Now comparing the exponents,

$\implies 4x<2$

$\implies \boxed{x<\dfrac{1}{2} }$

Hence, we proved $\dfrac{2}{5} <\log_{10}3<\dfrac{1}{2}$.

Answer 2

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