Solve 10^x - 0.47 * 10^x = 220
a. With an exact answer.
b. Decimal form with one decimal accuracy.
c. Decimal form with three values of accuracy.
10^x - 0.47 * 10^x = 220
0.53 * 10^x = 220
10^x = 220/0.53
x = log(220/0.53)
10^x - 0.47 * 10^x = 220
10^x[1-0.47]=220
10^x[0.53]=22×10
0.53=22×10^(1-x)
53×10^(-2)=22×10^(1-x)
53=[22×10^((1-x)-(-2)]
53=22×10^(3-x)
22×10^(3-x)=53
10^(3-x)=53/22
(3-x)log10=log53-log22
3-x=[log53-log22]/log10
3-x=1.7242-1.3424
3-x=0.3818
-x=0.3818-3
-x=−2.6182
x=2.6182
To solve the equation 10^x - 0.47 * 10^x = 220, we can follow these steps:
Step 1: Combine the like terms on the left side of the equation:
10^x - 0.47 * 10^x = 220
(1 - 0.47) * 10^x = 220
0.53 * 10^x = 220
Step 2: Divide both sides of the equation by 0.53 to isolate 10^x:
(0.53 * 10^x) / 0.53 = 220 / 0.53
10^x = 415.09433962
Now let's proceed to answer the given options:
a. With an exact answer:
To find the exact value of x, we can take the logarithm of both sides of the equation. Taking the base 10 logarithm will give us x.
log(base 10) (10^x) = log(base 10) 415.09433962
x * log(base 10) 10 = log(base 10) 415.09433962
x * 1 = log(base 10) 415.09433962
x = log(base 10) 415.09433962
The exact value of x is given by the logarithm of 415.09433962 to the base 10.
b. Decimal form with one decimal accuracy:
We can use the logarithm function on a scientific calculator or a math software to find the decimal value of x.
x ≈ 2.618
Rounding to one decimal accuracy, x ≈ 2.6.
c. Decimal form with three values of accuracy:
Using a calculator or math software, we can find the logarithm of 415.09433962 to the base 10, and round it to three decimal places.
x ≈ 2.618
Rounding to three decimal places, x ≈ 2.618.