INPUT ARTICLE: Article: You're probably really nervous about talking to your girlfriend, but it's the only way to know for sure if she's cheating. Explain that you care about your relationship and are worried that something is wrong. Then, say that you think she might be cheating and why you suspect that. You might say, "I really love you and want to have a future together. Lately, I've been worried that our relationship might be changing. I've noticed that you aren't as affectionate, disappear for hours at a time, and are spending more time on your appearance. These behaviors have me worried that you're cheating." She'll likely get upset when you bring up the topic of cheating, whether she is or isn't. Reassure her that you're trying to help your relationship and deserve the truth. Then, give her a chance to explain what's going on. You could say, "I see that you're upset, and that's not my intention. I want to make things better for us, but I need to know the truth." to her side of the story. There may be a perfectly valid reason for her behavior, so give her a chance to explain. Don't interrupt her while she's talking and encourage her to say all she needs to say. Additionally, paraphrase what she said so she knows you were listening. Say, "It sounds like you've been questioning our relationship," or "It sounds like you've just had more things on your plate than I realized." It's really hard to tell when someone is lying, but there are a few signs you can look for. First, consider how she normally acts in a conversation with you. Then, look for differences in the way she's talking to you when you ask her about cheating. Here are some indications that she might be telling a lie: Her body language suddenly changes when you ask her about cheating. She starts squirming, fidgeting, or pulling away. Her sentence structure changes to longer, more confusing sentences. She breaks eye contact. Her facial expression changes. She blushes, bites her lip, flares her nostrils, or sweats. She talks either faster or slower and her tone changes. Whether or not she admits to cheating, you need to figure out if this relationship is what's best for you. You deserve to feel secure in your relationship, so think about whether or not you feel like you can trust her. Then, decide if you want to work on the relationship or end it. If she admits to cheating, ask yourself if that's something you can get over. If it's not, it might be best to move on. Consider asking your girlfriend to work on things with you. If she shows an interest in saving your relationship, it might be worth trying to make things work.

SUMMARY: Tell your girlfriend about your suspicions. Expect her to get upset when you confront her about cheating. Listen Check for signs that she may be lying to you. Decide whether or not you're willing to continue the relationship.

INPUT ARTICLE: Article: This method covers problems in the form logb(x)logb(a){\displaystyle {
rac {\log _{b}(x)}{\log _{b}(a)}}}. However, it does not work for a few special cases: The log of a negative number is undefined for all bases (such as log(3){\displaystyle \log(-3)} or log4(5){\displaystyle \log _{4}(-5)}). Write "no solution." The log of zero is also undefined for all bases. If you see a term such as ln(0){\displaystyle \ln(0)}, write "no solution." The log of one in any base (log(1){\displaystyle \log(1)}) always equals zero, since x0=1{\displaystyle x^{0}=1} for all values of x. Replace that logarithm with 1 instead of using the method below. If the two logarithms have different bases, such as log3(x)log4(a){\displaystyle {
rac {log_{3}(x)}{log_{4}(a)}}}, and you cannot simplify either one into an integer, the problem is not feasible to solve by hand. Assuming you did not find any of the exceptions above, you can now simplify the problem into one logarithm. To do this, use the formula logb(x)logb(a)=loga(x){\displaystyle {
rac {\log _{b}(x)}{\log _{b}(a)}}=\log _{a}(x)}. Example 1: Solve the problem log16log2{\displaystyle {
rac {\log {16}}{\log {2}}}}.Start by converting this into one logarithm using the formula above: log16log2=log2(16){\displaystyle {
rac {\log {16}}{\log {2}}}=\log _{2}(16)}. This formula is the "change of base" formula, derived from basic logarithmic properties. Remember, to solve loga(x){\displaystyle \log _{a}(x)}, think "a?=x{\displaystyle a^{?}=x}" or "What exponent can I raise a by to get x?" It's not always feasible to solve this without a calculator, but if you're lucky, you'll end up with an easily simplified logarithm. Example 1 (cont.): Rewrite log2(16){\displaystyle \log _{2}(16)} as 2?=16{\displaystyle 2^{?}=16}. The value of "?" is the answer to the problem. You may need to find it by trial and error:22=22=4{\displaystyle 2^{2}=2*2=4}23=42=8{\displaystyle 2^{3}=4*2=8}24=82=16{\displaystyle 2^{4}=8*2=16}16 is what you were looking for, so log2(16){\displaystyle \log _{2}(16)} = 4. Some logarithms are very difficult to solve by hand. You'll need a calculator if you need the answer for a practical purpose. If you're solving problems in math class, your teacher most likely expects you to leave the answer as a logarithm. Here's another example using this method on a more difficult problem: Example 2: What is log3(58)log3(7){\displaystyle {
rac {\log _{3}(58)}{\log _{3}(7)}}}? Convert this into one logarithm: log3(58)log3(7)=log7(58){\displaystyle {
rac {\log _{3}(58)}{\log _{3}(7)}}=\log _{7}(58)}. (Notice that the 3 in each initial log disappears; this is true for any base.) Rewrite as 7?=58{\displaystyle 7^{?}=58} and test possible values of?:72=77=49{\displaystyle 7^{2}=7*7=49}73=497=343{\displaystyle 7^{3}=49*7=343}Since 58 falls between these two numbers, log7(58){\displaystyle \log _{7}(58)} has no integer answer. Leave your answer as log7(58){\displaystyle \log _{7}(58)}.

SUMMARY:
Check for negative numbers and ones. Convert the expression into one logarithm. Calculate by hand if possible. Leave the answer in logarithm form if you cannot simplify it.