log5 (8y - 6) - log5 (y - 5) = log4 16
To solve this equation, we can use logarithm properties. First, we can simplify the equation using the properties of logarithms.
Using the property log a - log b = log (a/b), we have:
log5(8y - 6) - log5(y - 5) = log4(16)
log5[(8y - 6)/(y - 5)] = log4(16)
Now, we can convert the equation into exponential form. Recall that log a b = x is equivalent to b = a^x.
From the equation above, we have:
(8y - 6)/(y - 5) = 16^4
Simplifying the right side, we have:
(8y - 6)/(y - 5) = 2^8
Next, we can solve this equation for y by cross-multiplying:
2^8(y - 5) = 8y - 6
Expanding both sides, we get:
256y - 1280 = 8y - 6
Combining like terms, we have:
248y - 1274 = 0
Adding 1274 to both sides:
248y = 1274
Dividing both sides by 248:
y = 1274/248
Simplifying the fraction, we get:
y = 5.1371
Therefore, the solution to the equation is y = 5.1371.
To solve the given equation step-by-step, let's break it down:
1. First, we can simplify the left side of the equation using the logarithmic properties. According to the property log base b (x) - log base b (y) = log base b (x/y), we can write:
log5 (8y - 6) - log5 (y - 5) = log5 [(8y - 6) / (y - 5)]
2. Next, we can simplify the right side of the equation. Since log4 16 is asking for the power to which we raise 4 to get 16, we can rewrite this as:
log4 16 = 2 (since 4^2 = 16)
3. Now our equation becomes:
log5 [(8y - 6) / (y - 5)] = 2
4. To eliminate the logarithm, we can rewrite the equation using the exponential form. If log base b (x) = y, then b^y = x. In this case, we have base 5 and exponent 2, so:
5^2 = (8y - 6) / (y - 5)
5. Simplifying 5^2 gives us:
25 = (8y - 6) / (y - 5)
6. To remove the fraction, we can multiply both sides of the equation by (y - 5):
25 * (y - 5) = 8y - 6
7. Expanding the left side:
25y - 125 = 8y - 6
8. Next, we can rearrange the equation to isolate the variable y on one side. Move 8y to the left side and add 125 to both sides of the equation:
25y - 8y = 125 - 6
17y = 119
9. Divide both sides of the equation by 17 to solve for y:
y = 119 / 17
10. Finally, simplify the fraction if possible:
y = 7
Therefore, the solution to the equation log5 (8y - 6) - log5 (y - 5) = log4 16 is y = 7.
To solve this equation, we will use the properties of logarithms.
First, we will simplify each side of the equation separately.
On the left side, we subtract the two logarithms:
log5 (8y - 6) - log5 (y - 5)
To subtract logarithms with the same base, we can rewrite it as the division of the two logarithms:
log5 ((8y - 6)/(y - 5))
On the right side, we have log4 16.
Since the logarithm on the right side has a base of 4, and we have logarithms with a base of 5 on the left side, we will need to convert the logarithm on the right side to a base of 5.
To convert a logarithm from one base to another, we can use the change of base formula:
log4 16 = log5 16 / log5 4
Now we can rewrite the equation as:
log5 ((8y - 6)/(y - 5)) = log5 16 / log5 4
Next, we will focus on the right side of the equation.
Using logarithmic properties, we know that log5 a / log5 b is equal to logb a.
So, we can simplify the equation to:
log5 ((8y - 6)/(y - 5)) = log4 16
Since log5 ((8y - 6)/(y - 5)) = log4 16, we can conclude that:
((8y - 6)/(y - 5)) = 16
Now, we have a simple algebraic equation to solve.
To do so, we will cross-multiply:
(8y - 6) = 16(y - 5)
Distribute on the right side:
8y - 6 = 16y - 80
Move the variables to one side and constants to the other side:
8y - 16y = -80 + 6
-8y = -74
Divide both sides by -8 to solve for y:
y = (-74)/(-8)
y = 9.25
Therefore, the solution to the equation is y = 9.25.