Find x such that log3(2x-1)=1+log2(x+1)?
log3(2x-1)=1+log2(x+1)
2x-1 = 3*3^log2(x+1)
This does not yield to algebraic solution
However, if you had a typo for
log3(2x-1)=1+log3(x+1)
2x-1 = 3(x+1)
x = -4
But again, that has no real solution, since the logs are undefined.
Assaiment
To solve the equation log3(2x-1) = 1 + log2(x+1), we need to apply properties of logarithms and algebraic manipulation. Here's how you can solve it step by step:
Step 1: Use the property log(a) + log(b) = log(a * b) to rewrite the equation:
log3(2x - 1) = log2(x + 1) + log2(2)
Step 2: Use the property log(a^n) = n * log(a) to simplify further:
log3(2x - 1) = log2[(x + 1) * 2]
Step 3: Apply the property loga(a) = 1 to simplify the right side:
log3(2x - 1) = log2(2x + 2)
Step 4: Since the bases of the logarithms are different, we can convert them to the same base. Let's use the base 10 logarithm (log10) as it is commonly used:
log3(2x - 1) = log2(2x + 2) can be rewritten as: log(2x - 1) / log(3) = log(2x + 2) / log(2)
Step 5: Use the property loga(b) = logc(b) / logc(a) to simplify:
(log(2x - 1) / log(3)) = (log(2x + 2) / log(2))
Step 6: Cross-multiply and eliminate the denominators:
(log(2x - 1) * log(2)) = (log(2x + 2) * log(3))
Step 7: Use the property logb(b^a) = a to simplify:
log(2x - 1) * log(2) = log(2x + 2) * log(3) can be rewritten as: log(2) * log(2x - 1) = log(3) * log(2x + 2)
Step 8: Simplify the equation further:
log^2(2) + log(2) * log(1 - 2x) = log(2) * log(3) + log(3) * log(2x + 2)
Step 9: Combine like terms on both sides of the equation:
log^2(2) + log(2) * log(1 - 2x) - log(2) * log(3) - log(3) * log(2x + 2) = 0
Step 10: Let's denote log(2) as a and log(3) as b for convenience. The equation becomes:
a^2 + a * log(1 - 2x) - a * b - b * log(2x + 2) = 0
Step 11: Solve the quadratic equation for a. Rearrange the equation as follows:
a^2 + a * (log(1 - 2x) - b) - b * log(2x + 2) = 0
Step 12: Use the quadratic formula to find a:
a = [-b ± sqrt((log(1 - 2x) - b)^2 + 4b * log(2x + 2))] / 2
Step 13: Substitute back log(2) for a and log(3) for b:
log(2) = [-log(3) ± sqrt((log(1 - 2x) - log(3))^2 + 4 * log(3) * log(2x + 2))] / 2
Step 14: Simplify the equation:
2 * log(2) = -log(3) ± sqrt((log(1 - 2x) - log(3))^2 + 4 * log(3) * log(2x + 2))
Step 15: Solve for x:
1 - 2x = 10^[(2 * log(2) + log(3) ± sqrt((log(1 - 2x) - log(3))^2 + 4 * log(3) * log(2x + 2))]/2
Step 16: Rearrange and simplify the equation to isolate x:
x = [1 - 10^[(2 * log(2) + log(3) ± sqrt((log(1 - 2x) - log(3))^2 + 4 * log(3) * log(2x + 2))]/2] / 2
Please note that this equation is quite complex, and it might not be possible to obtain an explicit solution for x. However, you can use numerical methods or approximation techniques to find an approximate value for x.