Question

Find the value of log√[5√{5√(5...infinite)}]/log5​

Answer 1

Answer:

1

Step-by-step explanation:

x = log√[5√{5√(5...)}] / log5​

=> x = log₅√[5√{5√(5...)}]

=> 5ˣ = √[5√{5√(5...)}]

=> 5²ˣ = 5√[5√{5√(5...)}]

=> 5²ˣ⁻¹ = √[5√{5√(5...)}]

=> 5²ˣ⁻¹ = 5ˣ

=> 2x - 1 = x

=> x = 1

Answer 2

Answer:

The value of log√[5√{5√(5...infinite)}]/log5​ = 1

Step-by-step explanation:

Suppose, x = log√[5√{5√(5...infinite)}]/log5​.........(1)

• Step - 1: From the formula of logarithm we know  $\frac{\log_{10}x}{\log_{10}a}$ = $\log_{a} x$

                      So, $\frac{\log_{10} \sqrt{5\sqrt{5\sqrt{5}.....infinite} } }{\log_{10}5}$ = ${\log_{5} \sqrt{5\sqrt{5\sqrt{5}.....infinite} } }$

• Step - 2:  From equation (1), we can write,

                        x = ${\log_{5} \sqrt{5\sqrt{5\sqrt{5}.....infinite} } }$.............(2)

• Step - 3: From the formula of logarithm, if y = $\log_{a} x$

                                                                      then, $a^y$ = x

    Applying this formula in equation (2) we get,

                  $5^x$ = ${\sqrt{5\sqrt{5\sqrt{5}.....infinite} } }$...................(3)

        [ taking square on both sides]

                 $5^(2x)$ = 5 ${\sqrt{5\sqrt{5\sqrt{5}.....infinite} } }$ ...............(4)

• Step - 4: Comparing (3) and (4), we get,

                                $5^(2x)$ = 5 ×  $5^x$

                                $\frac{5^2x}{5^x}$ = 5

                                $5^(2x-x)$ = 5

                                 $5^x$ = 5 = $5^1$

                            ∴ x = 1

• Conclusion: From equation (1) as x = log√[5√{5√(5...infinite)}]/log5

                                   and finally we get x = 1

    So we can say, log√[5√{5√(5...infinite)}]/log5 = 1