Find the value of X if;
Log base 10 7+ log base 10(3x+2)-log base 10(2x-1)=1
log [7(3x+2)/(2x-1)]= 1
then
[7(3x+2)/(2x-1)] = 10^1 = 10
7(3x+2) = 20x-10
21 x + 14 = 20 x - 10
etc
Well, well, well! It seems we have quite the mathematical conundrum here. Before we dive into the logarithmic humor, let's set the stage for our grand finale: finding the value of X.
Are you ready for some math magic? Here we go!
Let's start by simplifying the given equation:
log base 10(7) + log base 10(3x + 2) - log base 10(2x - 1) = 1
Ah, the symphony of logs and numbers. Now, buckle up and get ready to solve this equation like a true math magician!
To make our lives a bit easier, we can combine the logs using the properties of logarithms. The addition of logs is equivalent to multiplication.
log base 10(7 * (3x + 2) / (2x - 1)) = 1
Now, let's convert this equation back to exponential form.
10^1 = 7 * (3x + 2) / (2x - 1)
We can simplify further:
10 = 7 * (3x + 2) / (2x - 1)
Hang in there, we're almost there!
Next, let's cross-multiply to eliminate the fraction:
10 * (2x - 1) = 7 * (3x + 2)
Multiply each term:
20x - 10 = 21x + 14
Now, let's bring all the X's to one side and the numbers to the other:
20x - 21x = 14 + 10
-x = 24
And now, for the grand finale, solving for X:
x = -24
Ta-da! We found our answer, and it's X = -24. I hope you enjoyed this mathematical spectacle. Remember, whether positive, negative, or even imaginary, numbers always love a good laugh!
To solve the equation, we can use logarithmic properties to combine the terms. The logarithmic property we will use here is the quotient rule, which states that:
log(base a)(b) - log(base a)(c) = log(base a)(b/c)
Applying the quotient rule, the equation becomes:
log(base 10)(7) + log(base 10)(3x+2) - log(base 10)(2x-1) = 1
Using the sum rule of logarithms, we can combine the first two terms:
log(base 10)[7(3x+2)] - log(base 10)(2x-1) = 1
Now, we can rewrite the equation as:
log(base 10)[21x + 14] - log(base 10)(2x - 1) = 1
Using the quotient rule, we can simplify further:
log(base 10)[(21x + 14)/(2x - 1)] = 1
Since the logarithm is equal to 1, we know that the argument of the logarithm must be equal to 10^1, which is 10. So we can write:
(21x + 14)/(2x - 1) = 10
Now we can solve for x by cross-multiplying:
(21x + 14) = 10(2x - 1)
21x + 14 = 20x - 10
Subtract 20x from both sides:
21x - 20x + 14 = -10
x + 14 = -10
Subtract 14 from both sides:
x = -10 - 14
x = -24
Therefore, the value of x that satisfies the equation is -24.
To find the value of x in the given equation, we need to simplify the equation using logarithm properties. Remember the following properties:
1. log base a (mn) = log base a (m) + log base a (n).
2. log base a (m/n) = log base a (m) - log base a (n).
3. log base a (m^p) = p * log base a (m).
Let's simplify the equation step by step:
Log base 10 7 + log base 10 (3x + 2) - log base 10 (2x - 1) = 1
Using property 1, we can combine the first two logarithms:
Log base 10 (7 * (3x + 2)) - log base 10 (2x - 1) = 1
Now, using property 2, let's subtract the third logarithm:
Log base 10 [(7 * (3x + 2))/(2x - 1)] = 1
To get rid of the logarithm, we exponentiate both sides with base 10:
10^1 = (7 * (3x + 2))/(2x - 1)
Simplify:
10 = (7 * (3x + 2))/(2x - 1)
Now, let's cross-multiply:
10(2x - 1) = 7(3x + 2)
20x - 10 = 21x + 14
Rearrange the equation:
20x - 21x = 14 + 10
-x = 24
Divide both sides by -1:
x = -24
Therefore, the value of x is -24.