Question

How do you find the value of ${\log _{\dfrac{1}{2}}}15$ using the change of base formula?

Answer 1

Hint: We know that a formula that allows us to write a logarithm in terms of logs written with another base is called change of base formula. This is especially helpful when we have to evaluate log to any base other than $10$ or $e$ .
 We will use the formula of change of base formula which is given by
 ${\log _b}a = \dfrac{{{{\log }_c}a}}{{{{\log }_c}b}}$ .
Also we can see that the base is in fraction or division form, so we will use another division logarithm formula..

Complete step by step solution:
Here we have ${\log _{\dfrac{1}{2}}}15$
By comparing with the change of base formula we have
 $a = 15,b = \dfrac{1}{2}$
Now we can write it as
 $\dfrac{{\log 15}}{{\log \dfrac{1}{2}}}$
Again we will apply the formula in the denominator i.e.
 ${\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y$
By comparing we have
$x = 1,y = 2$
Now we will put the values, so we have
$\dfrac{{\log 15}}{{\log 1 - \log 2}}$
We will substitute the values of the logarithm i.e.
$\log 15 = 1.1761,\log 2 = 0.3010$
And the value of
 $\log 1 = 0$
Therefore we have
$\dfrac{{1.1761}}{{0 - 0.3010}}$
It gives us value $ - 3.9073$ .
Hence the required answer is $ - 3.9073$ .

Additional information:
We should know that the logarithm function is also defined by if
 ${\log _a}b = x$ , then ${a^x} = b$ .
Where $x$ is defined as the logarithm of a number “b” and “a” is the base function that can have any value but usually we consider it as $e$ or $10$ .

Note:
We should note that the value of “a” can be any positive number but not equal to $1$ or negative number.
We must know that
 $\log 1 = 0$ and $\log 10 = 1$ .
The logarithm function with base $10$ is called the common logarithmic functions and log with base $e$ is called the neutral logarithmic function.