(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.