log (x-9) - 109x=log 19

10th grade is NOT the School Subject. Math, perhaps? Please label carefully so the right teacher will read and answer your post.

Sra

log10

To solve the equation log (x-9) - 109x = log 19, we can use logarithmic properties and algebraic manipulations. Here's how you can approach it:

Step 1: Combine the logarithms on the left side of the equation using the properties of logarithms, specifically the subtraction property that states log(a) - log(b) = log(a/b). The equation becomes log((x-9)/19) - 109x = 0.

Step 2: To eliminate the logarithm, convert the equation to exponential form. In exponential form, log(base b) (a) = c can be written as b^c = a. Applying this to our equation, we have (x-9)/19 - 109x = 1.

Step 3: Simplify the equation by multiplying both sides by 19 to get rid of the fraction: x - 9 - 2071x = 19.

Step 4: Combine like terms by adding 9 and multiplying 2070 with 19 on both sides of the equation: -2052x = 19 + 9 - (19 * 2070).

Step 5: Continue simplifying the equation by performing the arithmetic: -2052x = 19 - 2070 + 17190.

Step 6: Further simplify and solve for x: -2052x = 15140.

Step 7: Divide both sides of the equation by -2052: x = 15140 / -2052.

Step 8: Finally, compute the result using a calculator or long division to get the approximate value of x. It turns out that x ≈ -7.37 (rounded to two decimal places).

Therefore, the solution to the equation log (x-9) - 109x = log 19 is x ≈ -7.37.