Express y as a function of x. What is the domain? Explain your answer.
log4y = x + log4
Solving exponential equations with logarithms
Solve to the nearest thousandth.
5^1+x = 2^1-x
If you mean
log(4y) = x + log(4)
Then you have
log(4y/4) = x
log(y) = x
x = 10^y
Assuming the usual lack of care with parentheses, I'll guess you mean
5^(1+x) = 2^(1-x)
5*5^x = 2 * 1/2^x
(5/2) = 1/(10^x)
10^x = 2/5
x = log(2/5)
To express y as a function of x, we can first rewrite the equation log4y = x + log4 using properties of logarithms.
By the rule of logarithmic identities, we can rewrite log4y as log4(y) = log4(4y/4) = log4(4y) - log4(4) = log4(4) + log4(y) - log4(4) = log4(4) + log4(y) - 1 = 1 + log4(y) - 1 = log4(y).
So, the equation becomes log4(y) = x + log4.
Next, we can raise both sides of the equation as exponents with base 4 to eliminate the logarithm.
4^(log4(y)) = 4^(x + log4).
Applying the property of logarithmic exponentiation (inverse of logarithmic function), we have y = 4^(x + log4) = 4^x * 4^(log4).
Since 4^(log4) equals 4 raised to the power that results in 4, it simplifies to 1. Therefore, we have y = 4^x * 1 = 4^x.
The domain is all real numbers since there are no restrictions on the values that x can take. In other words, x can be any real number.
To express y as a function of x in the equation log4y = x + log4, we can use the properties of logarithms.
First, we apply the property that states log(a) - log(b) = log(a/b). Applying this property to the equation, we can rewrite it as:
log4(y/4) = x
Next, we use the property that log(base a) of a value is equal to 1. In this case, log4(4) = 1. So, we can simplify the equation further:
log4(y/4) = x + 1
To eliminate the logarithm, we can use the property that states if log(base a) of b equals c, then a^c = b. We can rewrite the equation accordingly:
4^(x+1) = y/4
To simplify further, we can multiply both sides of the equation by 4:
4 * 4^(x+1) = y
Simplifying the equation, we get:
16 * 4^x = y
Now, we have expressed y as a function of x.
To find the domain of this function, we consider the restrictions on the base of the logarithm and any other restrictions in the equation. In this case, the base of the logarithm is 4, and since logarithms are defined for positive numbers greater than 0, the domain of the function is all real numbers, excluding when 4^x equals 0.
In summary, the equation expresses y as a function of x as y = 16 * 4^x, and the domain of this function is all real numbers except when 4^x equals 0.