Given that x=lg a is a solution of the equation 10^(2x+1)-7(10^x) =26. Find the value of a.
10 *(10^x)^2 - 7 (10^x) = 26
let z = 10^x
10 z^2 - 7 z -26 = 0
( z - 2)( 10 z + 13) = 0
10^x = 2 or 10^x = -1.3
if you log is base 10
log 10^x = x = log 2
but if x = log a
then log a = log 2
so a = 2
similarly
log 10^x = x = log (-1.3) but not allowed :)
To find the value of a, we need to substitute x = lg a into the equation and solve for a.
Let's start by substituting x = lg a into the given equation:
10^(2x + 1) - 7(10^x) = 26
Substituting x = lg a:
10^(2(lg a) + 1) - 7(10^(lg a)) = 26
Now, we need to simplify the equation. Let's simplify the exponents using the properties of logarithms:
Remember the property: 10^(lg a) = a
10^(2(lg a) + 1) - 7(10^(lg a)) = 26
Now, rewrite the equation:
10^(lg a)^2 * 10 - 7a = 26
Combining like terms:
10^(2lg a) * 10 - 7a = 26
Using the property: 10^(2lg a) = a^2
a^2 * 10 - 7a = 26
Now, we have a quadratic equation. Let's rearrange it to standard form:
a^2 * 10 - 7a - 26 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
Unfortunately, the quadratic equation does not factor easily, so we will use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 10, b = -7, and c = -26.
Substituting these values into the formula:
a = (-(-7) ± √((-7)^2 - 4(10)(-26))) / (2(10))
Simplifying:
a = (7 ± √(49 + 1040)) / 20
a = (7 ± √(1089)) / 20
Taking the square root of 1089:
a = (7 ± 33) / 20
Now, we have two possible solutions:
1. a = (7 + 33) / 20
a = 40 / 20
a = 2
2. a = (7 - 33) / 20
a = -26 / 20
a = -1.3
Therefore, the value of a can be either 2 or -1.3.