1. ## Step by step guiding required for logarithms sums

Hi,
I have encountered some challenging sums when doing my revision for my math test. Please try to help me as I will be doing the test tomorrow.

Here are the sums:

1)Find the exact value of x if (3x)^lg3=(4x)^lg4.

2)Without using a calculator, evaluate (lg 5)^2 + lg 2 lg 50.

3) Solve x in terms of a given that log base a 5 + 2= log base a (x+a) + log base a (x-3a).

Help would be appreciated. Thanks in advance.

2. Hello AeroScizor

Welcome to Math Help Forum!
Originally Posted by AeroScizor
Hi,
I have encountered some challenging sums when doing my revision for my math test. Please try to help me as I will be doing the test tomorrow.

Here are the sums:

1)Find the exact value of x if (3x)^lg3=(4x)^lg4.

2)Without using a calculator, evaluate (lg 5)^2 + lg 2 lg 50.

3) Solve x in terms of a given that log base a 5 + 2= log base a (x+a) + log base a (x-3a).

Help would be appreciated. Thanks in advance.
1) Take logs of both sides:
$\displaystyle \log3\log(3x)=\log4\log4x$

$\displaystyle \Rightarrow \log3(\log 3 + \log x)=\log4(\log4 + \log x)$
Solve this equation for $\displaystyle \log x$, using the 'difference of two squares' to simplify the result.

Answer: $\displaystyle x = \frac{1}{12}$

2) Note that $\displaystyle \log 2 = \log 10 - \log 5 = 1 - \log 5$, and that $\displaystyle \log50 = \log10+\log5 = 1+ \log 5$

So $\displaystyle \log2\log50 =$ ... (Use the difference of two squares again.)

Answer: $\displaystyle 1$

I'll come back to (3) later, when I have time, if no-one else has answered.

3. Hello again AeroScizor
Originally Posted by AeroScizor
...
3) Solve x in terms of a given that log base a 5 + 2= log base a (x+a) + log base a (x-3a).

Help would be appreciated. Thanks in advance.
Write $\displaystyle 2$ as $\displaystyle 2\log_a(a)=\log_a(a^2)$, and then:
$\displaystyle \log_a5+\log_a(a^2) = \log_a(x+a)+\log_a(x-3a)$

$\displaystyle \Rightarrow \log_a(5a^2)=\log_a(x+a)(x-3a)$

$\displaystyle \Rightarrow 5a^2 = ...$
Simplify, factorise and solve the quadratic in $\displaystyle x$. Not both roots are valid though, because $\displaystyle a,\; (x+a)$ and $\displaystyle (x-3a)$ must all be positive.

Answer: $\displaystyle x = 4a$

Can you complete it now?

Hello again AeroScizorWrite $\displaystyle 2$ as $\displaystyle 2\log_a(a)=\log_a(a^2)$, and then:
$\displaystyle \log_a5+\log_a(a^2) = \log_a(x+a)+\log_a(x-3a)$

$\displaystyle \Rightarrow \log_a(5a^2)=\log_a(x+a)(x-3a)$

$\displaystyle \Rightarrow 5a^2 = ...$
Simplify, factorise and solve the quadratic in $\displaystyle x$. Not both roots are valid though, because $\displaystyle a,\; (x+a)$ and $\displaystyle (x-3a)$ must all be positive.

Answer: $\displaystyle x = 4a$

Can you complete it now?