Log8 4 + log16 2 x log 27 81= x solve for x
To solve the equation, we will begin by applying the logarithmic properties to simplify the equation. The properties we will use are:
1. log_b (a) + log_b (c) = log_b (a * c)
2. log_b (a^n) = n * log_b (a)
3. log_b (b) = 1
Using these properties, let's simplify the equation step-by-step:
1. log8 4 + log16 2 x log27 81 = x
Using the property log_b (a) + log_b (c) = log_b (a * c), we can rewrite the equation as:
log8 (4 * 16 * 27) = x
2. log8 (4 * 16 * 27) = x
Simplifying within the logarithm, 4 * 16 * 27 equals 1728:
log8 1728 = x
3. log8 1728 = x
Using the property log_b (a^n) = n * log_b (a), we can rewrite the equation as:
3 * log8 2 = x
4. 3 * log8 2 = x
Now, we can simplify further by evaluating log8 2. Since 2^3 = 8, we know that log8 2 = 3:
3 * 3 = x
5. 3 * 3 = x
Simplifying the equation, we find that:
x = 9
Therefore, the solution to the equation log8 4 + log16 2 x log27 81 = x is x = 9.
To solve the equation log8 4 + log16 2 x log 27 81 = x, we will simplify each logarithmic expression and then perform the arithmetic operations.
First, let's simplify each logarithmic expression using the logarithmic properties:
1. log8 4: We can rewrite this logarithmic expression as log2 4 / log2 8 since loga b = logc b / logc a. In this case, since the base is 8, which can be rewritten as 2^3, the log base 2 of 8 is equal to 3. Therefore, log2 4 is 2 / 3.
2. log16 2: Similarly, we can rewrite this logarithmic expression as log2 2 / log2 16. The log base 2 of 16 is equal to 4 since 2^4 = 16. Thus, log16 2 simplifies to 1 / 4.
3. log27 81: We can rewrite this logarithmic expression as log3 81 / log3 27. The log base 3 of 27 is equal to 3 since 3^3 = 27. The log base 3 of 81 is equal to 4 since 3^4 = 81. Therefore, log27 81 simplifies to 4 / 3.
Now, let's substitute the simplified expressions back into the equation:
(x^2 / 3) + (1 / 4) * (4 / 3) = x
Next, let's simplify the right side of the equation:
(x^2 / 3) + (4 / 12) = x
(x^2 / 3) + (1 / 3) = x
Now, we can multiply both sides of the equation by 3 to eliminate the denominators:
x^2 + 1 = 3x
Rearranging the equation, we have:
x^2 - 3x + 1 = 0
Unfortunately, this is a quadratic equation. To solve it further, you can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -3, and c = 1. Plugging these values into the quadratic formula will yield the values of x.
Log8 4 = 2/3 since 8^2/3 = 4
Log16 2 = 1/4 since 16^1/2 = 2
Log27 81 = 4/3 since 27^4/3 = 81
The sum of those logs is x, which equals 6.25