(I have two questions)
1. Solve for x without a calculator:
log3(x^2)=2log3(4)-4log3(5)
This is what I have so far:
2log3(x)=log3(4^2)-log3(5^4)
2log3(x)=log3(16/625)
...and now I'm stuck.
2. Simplify:
2log4(9)-2log3 <--how do you simplify with different bases?
Thanks!
from
2log3(x)=log3(16/625)
log3 (x^2) = log3 (16/625
so x^2 = 16/625
x = √(16/625) = 4/25
1. To solve the equation log3(x^2) = 2log3(4) - 4log3(5), you should use the logarithmic rules and properties. Here's how you can continue from where you left off:
2log3(x) = log3(16/625)
First, use the power rule: logb(a^c) = clogb(a)
2log3(x) = log3(16) - log3(625)
Next, simplify the logarithmic expressions on the right side:
2log3(x) = log3(2^4) - log3(5^4)
Apply the power rule again:
2log3(x) = 4log3(2) - 4log3(5)
Now, use the subtraction rule: logb(a) - logb(c) = logb(a/c)
2log3(x) = log3(2^4/5^4)
Simplify further:
log3(x^2) = log3((16/625)^(1/2))
Apply the power rule to eliminate the logarithm:
x^2 = (16/625)^(1/2)
To solve for x, take the square root of both sides:
x = √(16/625)
Simplify to obtain the final value of x:
x = 4/25
Therefore, the solution to the equation is x = 4/25.
2. To simplify the expression 2log4(9) - 2log3, you need to work with the logarithmic rules for different bases. Here's how you can proceed:
Use the power rule to simplify the logarithmic terms:
2log4(9) - 2log3(1)
Since logx(1) = 0 for any base x, the second term becomes 0:
2log4(9) - 0
Next, apply the power rule to simplify the remaining logarithm:
log4(9^2)
Evaluate the exponent:
log4(81)
Simplify the logarithm using the change of base formula or common logarithm:
log4(81) = log(81) / log(4)
Use a calculator to find the logarithm values:
log4(81) ≈ 1.709
Therefore, the simplified expression is approximately equal to 2 * 1.709, which is approximately 3.418.
So, 2log4(9) - 2log3 simplifies to approximately 3.418.