Help with basic Logarithms

**BG5965** · November 15th 2008, 03:16 AM

Just a side question: Why is Log9(3) = 1/2? Can you please solve it step by step for me. Thanks

1) Solve

showing the steps.

2) Solve

showing all steps.

Thankyou!

**earboth** · November 15th 2008, 04:11 AM

Originally Posted by BG5965

Just a side question: Why is Log9(3) = 1/2? Can you please solve it step by step for me. Thanks

1) Solve

showing the steps.

2) Solve

showing all steps.

Thankyou!

To your first question: You are looking for an exponent which belongs to the base 9 such that the result is 3:

$9^x = 3~\implies~9^x = \sqrt{9} ~\implies~9^x = 9^{\frac12}$

2 powers with the same base are equal if the exponents are equal too. Therefore

$x = \dfrac12$

to #1:

$\dfrac{\log_7(32)}{\log_7(8\sqrt{2}}=\dfrac{5\log_ 7(2)}{3.5\log_7(2)}= \dfrac{10}7$

to #2:

$\dfrac{\log_a(18)+\log_a(12)-\log_a(125)}{\log_a(42) - \log_a(35)} =$ $\dfrac{2\log_a(3)+\log_a(2)+2\log_a(2)+\log_a(3)-3\log_a(5)}{\log_a(6)+ \log_a(7) - \log_a(7)-\log_a(5)} =$ $= \dfrac{3\left(\log_a(2)+\log_a(3)-\log_a(5) \right)}{2\log_a(7)+\log_a(6)-\log_a(5) }$

**Soroban** · November 15th 2008, 08:34 AM

Hello, BG5965!

$2)\;\;\frac{\log_a(18) + \log_a(12) - \log_a(125)}{\log_a(42) - \log_a(35)}$

The numerator is: . $\log_a(2\cdot3^2) + \log_a(2^2\cdot3) - \log_a(5^3)$

. . $= \;\log_a(2) + \log_a(3^2) + \log_a(2^2) + \log_a(3) - \log_a(5^3)$

. . $= \;\log_a(2) + 2\log_a(3) + 2\log_a(2) + \log_a(3) - 3\log_a(5)$

. . $= \;3\log_a(2) + 3\log_a(3) - 3\log_a(5)$

. . $= \;3\bigg[\log_a(2) + \log_a(3) - \log_a(5)\bigg]$

The denominator is: . $\log_a(2\cdot3\cdot7) - \log_a(5\cdot7)$

. . $= \;\bigg[\log_a(2) + \log_a(3) + \log_a(7)\bigg] - \bigg[\log_a(5) + \log_a(7)\bigg]$

. . $= \;\log_a(2) + \log_a(3) - \log_a(5)$

The fraction becomes: . $\frac{3\left[\log_a(2) + \log_a(3) - \log_a(5)\right]} {\log_a(2) + \log_a(3) - \log_a(5)} \;=\;\boxed{3}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An alternate method . . .

We have . $\frac{\log_a(18) + \log_a(12) - \log_a(15)}{\log_a(42) - \log_a(35)} \;= \;\frac{\log_a\left(\dfrac{18\cdot12}{125}\right)} {\log_a\left(\dfrac{42}{35}\right)}$

. . . . . . $= \;\dfrac{\log_a(1.728)}{\log_a(1.2)} \;=\;\log_{1.2}(1.728)$

Since $1.2^3 \:=\:1.728$ , then: . $\log_{1.2}(1.728) \;=\;3$

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