Question

To determine

(a) The value of \({\log _{12}}10\).

(b) The value of \({\log _2}8.4\).

Short answer

(a) The value of \({\log _{12}}10 \approx 0.926628\).

(b) The value of \({\log _2}8.4 \approx 3.070390\).

Step-by-step solution

Given data

The given functions are \({\log _{12}}10\) and \({\log _2}8.4\).

Concept of Logarithm

Logarithm is the exponent or power to which a base must be raised to yield a given number.

For all\(a > 0\)and\(a \ne 1\;,\;{\log _a}x = \frac{{\ln x}}{{\ln a}}\).

Calculation of the value of \({\log _{12}}10\)

(a)

Simplify the given quantity \({\log _{12}}10\) as follows.

\(\begin{array}{c}{\log _{12}}10 = \frac{{\ln 10}}{{\ln 12}}\\{\log _{12}}10 = \frac{{2.302585}}{{2.484907}}\\{\log _{12}}10 \approx 0.926628\end{array}\)

Thus, the value of \({\log _{12}}10 \approx 0.926628\).

Calculation of the value of \({\log _2}8.4\)

(b)

Simplify \({\log _2}8.4\) as shown below.

\(\begin{array}{c}{\log _2}8.4 = \frac{{\ln 8.4}}{{\ln 2}}\\{\log _2}8.4 = \frac{{2.128232}}{{0.693147}}\\{\log _2}8.4 \approx 3.070390\end{array}\)

Thus, the value of \({\log _2}8.4 \approx 3.070390\).