Solve the equation to find the value of x : log10 (3x+2)

assuming base 10 for the logs,

log(3x+2) – 2 log(x) = 1– log(5x – 3)
log(3x+2) – 2 log(x) + log(5x – 3) = 1
Now, using the basic properties of logs, we have
log (3x+2)(5x-3)/x^2 = 1
(3x+2)(5x-3) = 10x^2
(6x+5)(x-1) = 0
x = - 5/6 or 1
But logx is undefined for x<0, so x=1 is the only solution.
check:
log5 - 2log1 = 1-log2
log5+log2 = 1
log10 = 1
true!

log10 (2x + 5x _ 2)= 1

log10 (2x + 5x _ 2) = 1

To solve the given equation, we will first simplify it using logarithmic properties, and then apply algebraic techniques to isolate the variable x.

Step 1: Apply logarithmic properties
We can simplify the equation using the properties of logarithms, specifically the power rule and quotient rule.

Starting with the equation: log10(3x + 2) - 2log10(x) = 1 - log10(5x - 3)

Using the power rule of logarithms, we can rewrite 2log10(x) as log10(x^2):
log10(3x + 2) - log10(x^2) = 1 - log10(5x - 3)

Next, using the quotient rule of logarithms, we can rewrite log10(3x + 2) - log10(x^2) as a single logarithm:
log10((3x + 2)/(x^2)) = 1 - log10(5x - 3)

Step 2: Combine like terms
We can simplify further by rearranging the terms:
log10((3x + 2)/(x^2)) + log10(5x - 3) = 1

Step 3: Apply the product rule of logarithms
Using the product rule of logarithms, we can rewrite the left side of the equation as a single logarithm:
log10(((3x + 2)(5x - 3))/(x^2)) = 1

Step 4: Eliminate the logarithm
To eliminate the logarithm, we will rewrite the equation in exponential form. Since the base of the logarithm is 10, we can rewrite the equation as:
10^1 = ((3x + 2)(5x - 3))/(x^2)

Simplifying and removing the denominator:
10 = (3x + 2)(5x - 3)

Step 5: Expand and solve the quadratic equation
We can expand the expression on the right side of the equation:
10 = 15x^2 - 9x + 10x - 6

Combining like terms and rearranging:
0 = 15x^2 + x - 4

Next, we can either factor the quadratic equation or use the quadratic formula to solve for x. In this case, the equation does not factor nicely, so we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 15, b = 1, and c = -4. Substituting these values into the quadratic formula:

x = (-(1) ± √((1)^2 - 4(15)(-4))) / (2(15))

Simplifying further:

x = (-1 ± √(1 + 240)) / (30)
x = (-1 ± √241) / 30

Hence, the solution to the equation is:
x ≈ (-1 + √241) / 30 or x ≈ (-1 - √241) / 30