Q.Express the following in the alternative form:

The definition $a^{x}=N \Leftrightarrow \log_{a}N=x$ lets any statement be rewritten in the other form by identifying the base, the exponent and the number.

In (i) the base 2, exponent 5 and result 32 become $\log_{2}32=5$; in (ii) the base 3, logarithm 4 and number 81 become $3^{4}=81$.

The governing definition is

(i) Convert $2^{5}=32$ to logarithmic form. Here the base is $a=2$, the exponent is $x=5$, and the number is $N=32$. Substituting into $\log_{a}N = x$:

(ii) Convert $\log_{3}81 = 4$ to exponential form. Here the base is $a=3$, the logarithm (exponent) is $x=4$, and the number is $N=81$. Substituting into $a^{x}=N$:

Verification. (i) Check the exponential statement is genuinely true: $2^{5} = 2\times2\times2\times2\times2 = 32$ ✓, so its logarithmic form is valid. (ii) Check the other way: $3^{4} = 3\times3\times3\times3 = 81$ ✓, confirming $\log_{3}81 = 4$ was correctly re-expressed.

The commonest error is misplacing the parts — writing $\log_{32}2 = 5$ instead of $\log_{2}32 = 5$ by confusing the base with the number. The base of the power is always the base of the logarithm, and the result of the power is always the number whose logarithm is taken.