Question:
If
\[
\log_a 2+\log_a 5=1,
\]
then the value of \(a\) is:
If
\[
\log_a 2+\log_a 5=1,
\]
then the value of \(a\) is:
Show Hint
If
\[
\log_a b=1,
\]
then immediately conclude that
\[
a=b,
\]
because
\[
a^1=b.
\]
This shortcut frequently appears in logarithmic equations.
Updated On: Jun 10, 2026
- \(7\)
- \(10\)
- \(12\)
- \(15\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Logarithms obey several important laws which simplify complicated expressions. One of the most frequently used logarithmic identities is
\[
\log_a m+\log_a n=\log_a(mn).
\]
This property allows us to combine two logarithmic terms having the same base into a single logarithm. After combining the logarithms, we can convert the logarithmic equation into exponential form and solve it directly.
Whenever a logarithmic equation contains only one logarithm after simplification, converting to exponential form is generally the most efficient method.
Step 1: Write the given equation. We are given \[ \log_a2+\log_a5=1. \]
Step 2: Apply the logarithmic addition property. Using \[ \log_a m+\log_a n=\log_a(mn), \] we obtain \[ \log_a(2\times5)=1. \] Hence, \[ \log_a(10)=1. \]
Step 3: Convert the logarithmic equation into exponential form. The statement \[ \log_a(10)=1 \] means \[ a^1=10. \]
Step 4: Simplify. Since any number raised to the power \(1\) remains unchanged, \[ a=10. \]
Step 5: Verification. Substituting \(a=10\), \[ \log_{10}2+\log_{10}5 = \log_{10}(10) = 1. \] The given condition is satisfied.
Step 6: Final Conclusion. Therefore, \[ \boxed{a=10} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
Step 1: Write the given equation. We are given \[ \log_a2+\log_a5=1. \]
Step 2: Apply the logarithmic addition property. Using \[ \log_a m+\log_a n=\log_a(mn), \] we obtain \[ \log_a(2\times5)=1. \] Hence, \[ \log_a(10)=1. \]
Step 3: Convert the logarithmic equation into exponential form. The statement \[ \log_a(10)=1 \] means \[ a^1=10. \]
Step 4: Simplify. Since any number raised to the power \(1\) remains unchanged, \[ a=10. \]
Step 5: Verification. Substituting \(a=10\), \[ \log_{10}2+\log_{10}5 = \log_{10}(10) = 1. \] The given condition is satisfied.
Step 6: Final Conclusion. Therefore, \[ \boxed{a=10} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Exponential and Logarithmic Functions Questions
- The number of positive real roots of the equation \[ 3^{x+1} + 3^{-x+1} = 10 \] is:
- If $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$, then the value of $x$ is:
- The range of the function \( f(x) = \frac{x^2 + x + 1}{x^2 - x + 1} \) is:
- The remainder when the polynomial \( 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) is divided by \( x^2 - x - 3 \) is:
- The sum of the real roots of the equation $|x - 2|^2 + |x - 2| - 2 = 0$ is
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions