Question

How do you use ${\log _{10}}5 = .6990$ and ${\log _{10}}7 = .8451$ to evaluate the expression ${\log _{10}}35$ ?

Answer 1

Hint: We have to first convert the expression into the index form. We know that a power is the sum of a set of variables, all of which are identical. For example, the number ${3^7}$ is a power, with the number $3$ as the base and the number $7$ as the index or exponent.

Complete step by step answer:
We know that if ${a^b} = c$ then ${\log _a}c = b$
We have to begin solving, by first converting it from log form to the index form
$
  {\log _{10}}5 = 0.6990 \Rightarrow {10^{0.6990}} = 5 \\
  {\log _{10}}7 = 0.8451 \Rightarrow {10^{0.8451}} = 7 \\
 $
We know that the product of $5$ and $7$ is $35$
$35 = 5 \times 7$
Now we replace the values of $5$ and $7$with their equivalent index values and so we get,
$35 = {10^{0.6990}} \times {10^{0.8451}}$
Since the bases of both the values are the same $(10)$ and we are multiplying it, so we can add the indices.
We get,
$35 = {10^{1.5441}}$
We have found the index form. We have to convert the expression back to the log form.
Therefore, we get,
${\log _{10}}35 = 1.5441$

Note: Logarithms have been used in a variety of situations. To begin, logarithms are used to describe decibels, which are used to measure sound intensity. Second, logarithms are used to describe the Richter scale, which is used to calculate earthquake strength. Finally, the concept of a logarithm is used to describe the $pH$ value in chemistry, which is used to define the degree of acidity of a drug.
When two calculated quantities tend to be connected by an exponential function, log plots can be used to estimate the function's parameters. In experimental research, this is a very valuable instrument.