Question

How do you solve the equation $0.25 = \log {16^x}?$

Answer 1

Hint: Here first of all we will frame the given expression in the form of the required value and then use the logarithmic table for the reference value and then simplify it for the required resultant value.

Complete step-by-step solution:
Take the given expression: $0.25 = \log {16^x}$
 Apply Power rule: ${\log _a}{x^n} = n{\log _a}x$
$\Rightarrow 0.25 = x\log 16$
The above expression can be re-written as –
$\Rightarrow x\log 16 = 0.25$
Make the required variable “x” the subject and move other term on the opposite side. Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$\Rightarrow x = \dfrac{{0.25}}{{\log 16}}$
Refer, natural logarithmic table for the reference value for log
$\Rightarrow \log 16 = 1.2019$
Place it in the above equation –
$\Rightarrow x = \dfrac{{0.25}}{{1.2019}}$
Simplify the above expression –
$\Rightarrow x = 0.208$
This is the required solution.

Additional Information: Also refer to the below properties and rules of the logarithm.
Product rule: ${\log _a}xy = {\log _a}x + {\log _a}y$
Quotient rule: ${\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y$
Power rule: ${\log _a}{x^n} = n{\log _a}x$
 Base rule:${\log _a}a = 1$
Change of base rule: ${\log _a}M = \dfrac{{\log M}}{{\log N}}$

Note: Log table is used to find the value of the logarithmic function. Before the invention of the computers and other electronic devices, this log book was widely used and the easier way to get the required answer. In other words, the logarithm is the power to which the number must be raised in order to get some other. Always remember the standard properties of the logarithm.... Product rule, quotient rule and the power rule. The basic logarithm properties are most important and solution solely depends on it, so remember and understand its application properly. Be good in multiples and know the concepts of square and square root and apply accordingly.