Solve the equation=log _7 x+log_7 (18x+61)=1
To solve the equation, we can use the properties of logarithms.
First, we can combine the terms inside the logarithms using the product rule:
log_7(x) + log_7(18x + 61) = 1
Next, we can combine the logarithms into a single logarithm using the product rule:
log_7(x(18x + 61)) = 1
Now, we can rewrite the equation in exponential form:
7^1 = x(18x + 61)
Simplifying the left side:
7 = 18x^2 + 61x
Rearranging the equation:
18x^2 + 61x - 7 = 0
This is a quadratic equation. We can solve it by factoring or by using the quadratic formula. Since factoring is unlikely to yield rational roots for this equation, we will use the quadratic formula.
The quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 18, b = 61, and c = -7.
Plugging these values into the quadratic formula:
x = (-61 ± sqrt(61^2 - 4(18)(-7))) / (2(18))
Simplifying:
x = (-61 ± sqrt(3721 + 504)) / 36
x = (-61 ± sqrt(4225)) / 36
x = (-61 ± 65) / 36
This gives two possible solutions:
x1 = (-61 + 65) / 36 = 4 / 36 = 1/9
x2 = (-61 - 65) / 36 = -126 / 36 = -7/2
Therefore, the solutions to the equation are x = 1/9 and x = -7/2.
To solve the equation log_7 x + log_7 (18x + 61) = 1, we can use logarithmic properties to simplify the expression:
1. Combine the two logarithms using the product rule: log_7 (x * (18x + 61)) = 1
2. Rewrite the equation as an exponent: 7^1 = x * (18x + 61)
3. Simplify the left side of the equation: 7 = 18x^2 + 61x
4. Rearrange the equation to form a quadratic equation: 18x^2 + 61x - 7 = 0
Now, we can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring might be a little bit complicated, so we will use the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In the equation 18x^2 + 61x - 7 = 0, we have:
a = 18
b = 61
c = -7
Plugging the values into the quadratic formula, we get:
x = (-61 ± √(61^2 - 4 * 18 * -7)) / (2 * 18)
Simplifying the expression further:
x = (-61 ± √(3721 + 504)) / 36
x = (-61 ± √(4225)) / 36
x = (-61 ± 65) / 36
Now, we have two possible solutions for x:
1. x = (-61 + 65) / 36 = 4 / 36 = 1/9
2. x = (-61 - 65) / 36 = -126 / 36 = -7/2
Therefore, the solution to the equation log_7 x + log_7 (18x + 61) = 1 is x = 1/9 or x = -7/2.