solve log(7x-3)+2log(5)=2+log(x+3)
I've attempted to do this question and I ended up with log(7-3)+log(5^2)-log(x+3)=2 but I don't what to do next or whether I did something wrong.
write is as
log((7x-3)(25)/(x+3) = 2
25(7x - 3)/(x+3) = 10^2 = 100
divide by 25
(7x-3)(x+3) = 4
cross-multiply
7x-3 = 4x+12
3x = 15
x = 5
check:
LS = log 32 + log 25 - log 8
= log(32(25)/8)
= log 100
= 2
= RS
Just wondering what did you do in this log((7x-3)(25)/(x+3) = 2 to get 25(7x - 3)/(x+3) = 10^2 = 100?
To solve the given equation, let's go step-by-step.
Step 1: Apply the log properties to simplify the equation.
log(7x-3) + 2log(5) = 2 + log(x+3)
Using the log property log(a^n) = n*log(a), we can rewrite the equation as:
log(7x-3) + log(5^2) = 2 + log(x+3)
Simplifying further, we have:
log(7x-3) + log(25) = 2 + log(x+3)
Step 2: Combine the logarithms using the log property log(a) + log(b) = log(ab).
log((7x-3) * 25) = 2 + log(x+3)
Simplifying further, we have:
log(175x - 75) = 2 + log(x+3)
Step 3: Apply another log property, log(a) + log(b) = log(ab), to rewrite the equation as a single logarithm.
log(175x - 75) - log((x+3)) = 2
Step 4: Apply the log property log(a) - log(b) = log(a/b), to rewrite the equation as:
log((175x - 75)/(x+3)) = 2
Step 5: Convert the logarithmic equation to exponential form.
(175x - 75)/(x+3) = 10^2
Simplifying, we have:
(175x - 75)/(x+3) = 100
Step 6: Cross-multiply and solve for x.
175x - 75 = 100(x + 3)
175x - 75 = 100x + 300
Collecting like terms, we have:
175x - 100x = 300 + 75
75x = 375
Divide both sides by 75:
x = 375/75
x = 5
Therefore, the solution to the given equation is x = 5.
To solve the equation log(7x-3)+2log(5)=2+log(x+3), we can start by using logarithmic properties to simplify the equation.
First, let's apply the power rule of logarithms to the second term on the left side: 2log(5) = log(5^2).
Now, the equation becomes:
log(7x-3) + log(5^2) = 2 + log(x+3).
Next, we can use the product rule of logarithms to combine the terms on the left side:
log((7x-3) * 5^2) = 2 + log(x+3).
Simplifying further using the power rule: 5^2 = 25:
log(25 * (7x-3)) = 2 + log(x+3).
Applying the sum rule of logarithms to combine the terms on the right side:
log(25 * (7x-3)) = log((x+3) * 10^2).
Since the logarithm functions on both sides have the same base, we can drop the logarithm symbols:
25 * (7x-3) = (x+3) * 10^2.
Expanding and rearranging the equation:
175x - 75 = 100x + 300.
Bringing similar terms to one side:
175x - 100x = 300 + 75,
75x = 375.
Dividing both sides by 75:
x = 375 / 75,
x = 5.
So, the solution to the equation log(7x-3) + 2log(5) = 2 + log(x+3) is x = 5.
It's always important to check your answer, so take that x value and substitute it back into the original equation to verify if it satisfies the equation.