Question

If log /8 x = 3^1/3, find the value of x.​

Answer 1

Answer:

  The required solution is x = 1024

Solution:

Given, log₈x = 3 1/3

    or, log₈x = 10 / 3

    or, logₑx / logₑ8 = 10 / 3

    or, logₑx = (10 / 3) * logₑ8

    or, logₑx = (10 / 3) * logₑ(2³)

    or, logₑx = (10 / 3) * 3 * logₑ2

    or, logₑx = 10 * logₑ2

    or, logₑx = logₑ(2¹⁰)

    or, x = 2¹⁰

    or, x = 1024

∴ the required solution is x = 1024

Rules:

  • logₐb = logᵣb / logᵣa

  • log(aᵇ) = b * loga

  • logₛa = logₛb gives a = b

# giving another solution with another question:

$\displaystyle \mathsf{Now,\:log_{\sqrt{8}}x=3\frac{1}{3}}$

$\displaystyle \mathsf{or,\:\frac{log_{e}x}{log_{e}\sqrt{8}}=\frac{10}{3}}$

$\mathsf{or,\:log_{e}x=\frac{10}{3}\times log_{e}\sqrt{8}}$

$\displaystyle \mathsf{or,\:log_{e}x=\frac{10}{3}\times log_{e}(2^{\frac{3}{2}})}$

$\displaystyle \mathsf{or,\:log_{e}x=\frac{10}{3}\times \frac{3}{2}\times log_{e}2}$

$\displaystyle \mathsf{or,\:log_{e}x=5\times log_{e}2}$

$\displaystyle \mathsf{or,\:log_{e}x=log_{e}(2^{5})}$

$\displaystyle \mathsf{or,\:x=2^{5}=32}$

∴ the required solution is x = 32

Answer 2

Answer:

Step-by-step explanation:

Given, log₈x = 3 1/3

   or, log₈x = 10 / 3

   or, logₑx / logₑ8 = 10 / 3

   or, logₑx = (10 / 3) * logₑ8

   or, logₑx = (10 / 3) * logₑ(2³)

   or, logₑx = (10 / 3) * 3 * logₑ2

   or, logₑx = 10 * logₑ2

   or, logₑx = logₑ(2¹⁰)

   or, x = 2¹⁰

   or, x = 1024