(iii)   The Power Rule

When an exponential expression is raised to a power, we multiply the exponents i.e;(a^m)ⁿ = a^m.n

This property suggests the power rule.

Theorem: (Power Rule) Let a and x be positive real numebrs with a is not equal to 0 and n be any real number

then , log_aXⁿ = nlog_ax

i.e. the logarithm of a number with an exponent is the product of the exponent and the logerithm of that number

Can we find the value of x such that 2^x = 3⁵?In such cases we find the value of 3⁵ = 243.Then we can evaluate the value of x, for which the value of 2^xequals to 243.

Applying the logarithm and using the formula log_axⁿ = nlog_ax,Easily we can find the values of 3²⁵,3³³etc..

writing in logarthmic form

log₂3⁵ = x

5 log₂3 = x  (Therefore ; log_axⁿ = nlog_ax)

We observe that the value of x is the product of 5 and the value of log₂ 3.