The value of logsqrt(a) ( sqrt(a) ) is

Option c: 15/8. Solution: Let us first simplify the nested radical term: x = sqrt(a) Expressing the square roots as fractional exponents: x = a1/2 × (a × (a × a1/2)1/2)1/2 x = a1/2 × a1/4 × a1/8 × a1/16 Using the laws of exponents ap × aq = ap+q : x = a1/2 + 1/4 + 1/8 + 1/16 Calculating the sum in the exponent: 1/2 + 1/4 + 1/8 + 1/16 = 8 + 4 + 2 + 1/16 = 15/16 So, we have: x = a15/16 Now we need to evaluate the logarithm: logsqrt(a) ( sqrt(a) ) = loga1/2 ( a15/16 ) Using the base-power formula of logarithms logbq(ap) = p/q logb(a) : loga1/2 ( a15/16 ) = 15/16/1/2 loga(a) = 15/16 × 2 × 1 = 15/8 Hence, Option C is the correct answer.

Ratio, Proportion, Indices, LogarithmsPYQ Sept 25Question 4402 of 220

All Questions

The value of $\displaystyle \log_{\sqrt{a}} \left( \sqrt{a\sqrt{a\sqrt{a\sqrt{a}}}} \right)$ is

Options

\displaystyle \frac{7}{8}

\displaystyle \frac{15}{16}

\displaystyle \frac{15}{8}

\displaystyle \frac{3}{4}

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Correct Answer

✅ Option c — $\displaystyle \frac{15}{8}$

All Options:

A $\displaystyle \frac{7}{8}$
B $\displaystyle \frac{15}{16}$
C $\displaystyle \frac{15}{8}$
D $\displaystyle \frac{3}{4}$

Detailed Solution & Explanation

Let us first simplify the nested radical term:

x = \sqrt{a\sqrt{a\sqrt{a\sqrt{a}}}}

Expressing the square roots as fractional exponents:

x = a^{1/2} \cdot \left(a \cdot \left(a \cdot a^{1/2}\right)^{1/2}\right)^{1/2}

x = a^{1/2} \cdot a^{1/4} \cdot a^{1/8} \cdot a^{1/16}

Using the laws of exponents

\displaystyle a^p \cdot a^q = a^{p+q}

x = a^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}}

Calculating the sum in the exponent:

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = \frac{8 + 4 + 2 + 1}{16} = \frac{15}{16}

So, we have:

x = a^{15/16}

Now we need to evaluate the logarithm:

\log_{\sqrt{a}} \left( \sqrt{a\sqrt{a\sqrt{a\sqrt{a}}}} \right) = \log_{a^{1/2}} \left( a^{15/16} \right)

Using the base-power formula of logarithms

\displaystyle \log_{b^q}(a^p) = \frac{p}{q} \log_b(a)

\log_{a^{1/2}} \left( a^{15/16} \right) = \frac{15/16}{1/2} \log_a(a) = \frac{15}{16} \times 2 \times 1 = \frac{15}{8}

Hence, **Option C** is the correct answer.

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