1.evaluate without using a calculator
2logx+1=log900
2.solve the following equation
A. 3log(x-2)=4.714
B. 16^x-3 = 8^2x+1
C.10^2x=1/100
2logx + 1 = log900
since 1 = log10,
2logx + log10 = log900
log x^2 + log10 = log900
log(10x^2) = log 900
10x^2 = 900
x^2 = 90
x = √90
3log(x-2) = 4.714
log(x-2)^3 = 4.714
(x-2)^3 = 51760.68
x-2 = 37.268
x = 39.268
16^(x-3) = 8^(2x+1)
since 16 = 2^4 and 8 = 2^3,
2^(4(x-3)) = 2^(3(2x+1))
4x-12 = 6x+3
2x = -15
x = -15/2
10^2x = 1/100
10^2x = 10^-2
2x = -2
x = -1
To evaluate the equation 2logx+1=log900 without using a calculator, we need to use logarithmic properties and solve it step by step.
Step 1: Simplify the equation using log properties.
2logx + 1 = log900 can be rewritten as log(x^2) + log10 = log900.
Using the property loga + logb = log(ab), we can combine the logarithms: log(10x^2) = log900.
Step 2: Set the bases of the logarithms equal to each other.
10x^2 = 900.
Step 3: Solve for x.
Divide both sides of the equation by 10:
x^2 = 90.
Step 4: Take the square root of both sides.
x = ± √90.
Simplifying further, we have:
x ≈ ± 9.4868.
Thus, the solution for the equation 2logx+1=log900 is x ≈ ± 9.4868.
Now let's solve the following equations without using a calculator:
A. 3log(x-2) = 4.714.
Step 1: Divide both sides of the equation by 3.
log(x-2) = 4.714 / 3.
Step 2: Rewrite the equation in exponential form.
x-2 = 10^(4.714/3).
Step 3: Solve for x.
x = 2 + 10^(4.714/3).
B. 16^x-3 = 8^2x+1.
Step 1: Rewrite 16 and 8 in terms of powers of 2.
(2^4)^(x-3) = (2^3)^(2x+1).
Step 2: Apply the power rule of exponents.
2^(4(x-3)) = 2^(3(2x+1)).
Step 3: Since the bases are equal, set the exponents equal to each other.
4(x-3) = 3(2x+1).
Step 4: Solve for x.
Simplify the equation and solve:
4x - 12 = 6x + 3,
-2x = 15,
x = -15/2.
C. 10^(2x) = 1/100.
Step 1: Rewrite 1/100 as a power of 10.
10^(2x) = 10^-2.
Step 2: Since the bases are equal, set the exponents equal to each other.
2x = -2.
Step 3: Solve for x.
Divide both sides by 2:
x = -2/2,
x = -1.
Thus, the solutions for the equations are:
A. x = 2 + 10^(4.714/3).
B. x = -15/2.
C. x = -1.