Fundamental Theorem Of Calculus

To find the value of x given the equation log (x + 1) = 2,
we use the definition of logarithms.
Step 1: Apply the logarithm definition
The equation log (x + 1) = 2 means that 10 raised to the power of 2
equals x + 1:
Step 2: Solve for x
Calculate :
Step 3: Verify the solution
Substitute x = 99 back into the original equation:
Since ,
The equation holds true.
Step 4: Compare with options
- (a) 99 matches the calculated value.
- (b) 100 would give log .
- (c) 101 would give log .
- (d) 99.9 would give log .
Step 5: Conclusion
The value of x is 99, making (a) 99 the correct answer.

To solve the equation , we start by using the property of logarithms that states . Thus, we can rewrite the given equation as:

This can be further simplified by realizing that implies . Therefore:

Next, expand the left-hand side:

Equating both sides, we have:

Subtract 8 from both sides to set the quadratic equation to zero:

Now, solve this quadratic equation by factorization. We need two numbers whose product is and sum is . These numbers are and . Thus, we can factor the quadratic as:

Setting each factor equal to zero gives the solutions:

or

which implies:

or

However, since logarithms are only defined for positive arguments, we must ensure that the values of satisfy and which means . Therefore, is not valid. So, the only valid solution is .