Question

Log0.4(4x+10)-log0.4 1/5=log0.4 8

Answer 1

Log0.4 (4x+10)/5=log0.4 8
(4x+10)/5= 8
4x+10=40
4x=30
x= 7.5

Answer 2

To solve the logarithmic equation $\log_{0.4}(4x+10) - \log_{0.4}\left(\frac{1}{5}\right) = \log_{0.4}(8)$, you can use logarithm properties, specifically the quotient rule and the fact that $\log_a(b) = c$ is equivalent to $a^c = b$.

Here are the steps to solve it:

1. Apply the quotient rule, which states that $\log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right)$:

   $\log_{0.4}\left(\frac{4x+10}{\frac{1}{5}}\right) = \log_{0.4}(8)$

2. Now, we have an equation with the same base ($0.4$) on both sides. You can equate the arguments inside the logarithms:

   $\frac{4x+10}{\frac{1}{5}} = 8$

3. To simplify further, multiply both sides of the equation by $\frac{1}{5}$ to get rid of the fraction on the left side:

   $5(4x+10) = 8$

4. Distribute the 5 on the left side:

   $20x + 50 = 8$

5. Now, isolate the variable $x$:

   $20x = 8 - 50$ $20x = -42$

6. Finally, divide both sides by 20 to solve for $x$:

   $x = \frac{-42}{20}$ $x = -\frac{21}{10}$

So, the solution to the equation $\log_{0.4}(4x+10) - \log_{0.4}\left(\frac{1}{5}\right) = \log_{0.4}(8)$ is $x = -\frac{21}{10}$.